The Rule of Three (s) and the Probability of Nothing

A Re-edit and Posting of a 2008 Blog

From 1827 Pike's Arithmetic

In my youth, back when dinosaurs roamed the earth, there was “the rule of three”… singular, one, and even then the name was often described as “archaic”. More modern books tended to develop “properties of proportions” or similar terms for the problems of proportionalities. Now there seem to be an abundance of them; including one for witches, and one about businesses. There is not space enough to talk about all of them so I will mention three, of course.
The first rule of three is as old as math, and shows up at least as early as the Hindu mathematician Brahmagupta, and in Fibonacci’s famous Liber Abaci(1202). It was once so common that it was introduced into common language. Abraham Lincoln is quoted in his biography as stating that he learned to "read, write, and cipher to the rule of 3."   So common that student's often wrote verse like the following, in their copy (practice) books.

Multiplication is vexation;

Division is as bad;

The Rule of Three doth puzzle me,

And Practice drives me mad

The most common and longest living form was the direct rule (although there was an inverse rule as well), in which case three numbers would be given and a fourth sought so that the ratio between the third and fourth would match the ratio between the first and second; a:b = c:d. Today students use the ideas in elementary school to complete fraction equivalences, “2/3 is the same as 10/?” Some of the ancient examples grew incredibly complicated.

I suppose the reason I chose to address three of the many “rules of three” is because of the rule of three from language and literature. Three just seems to be the right number for lots of things, there were Three Musketeers, Three Stooges, and Three Coins in the Fountain. It was Goldilocks and the Three Bears, and “bah bah black sheep” had “three bags full.” Comics in the newspaper usually have three panels and many jokes involve a three part ritual where the punch line is the third element, such as the t-shirt with “Great Cities of the World” on the top, and below, one after another, “Paris, Rome, Fargo”. The first two make the last funnier. In language the examples range from “Blood, sweat, and tears, to vidi, vidi, vici. If you don’t think there really is a mental tendency to have three terms, consider that in Churchill’s speech, he actually used four; “I say to the House as I said to ministers who have joined this government, I have nothing to offer but blood, toil, tears, and sweat. “

The final rule of three I would mention is from statistics, and is of more recent origin. It is also, I think, a really clever solution to what is a really difficult problem. Suppose something never happens; how can you assign a probability to it? It is not that it might not happen some day, just not so far. It is just such a problem the statistical rule of there was created to handle. Suppose you stopped at the same gum ball machine every day, but unlike the normal gumball machine, this one did not have a glass you could see into the gumballs inside. You buy a gum ball every day and get red ones, and green ones, but never a blue one. After a while you begin to wonder if they even put a blue one in the machine. So one day, after 20 days of getting all the other colors, over lunch you ask your local statistician (doesn’t everyone have lunch with a statistician?) how to figure out if there really is a blue one in there. He pauses, fork poised in mid-air, and informs you that you can be 95% sure (a common statistical benchmark) that the proportion of blue gum balls is no greater than 14.3%. He had mentally taken three, and divided by one more than the number of failed efforts, to get 3/21 or 1/7 as the upper limit of the possible fraction.
The idea is base on a simple extension of the binomial probability. If you knew that P % of the gum balls were blue, then you could calculate the probability that None showed up in 20 days. The probability would be (1-p)²⁰. Working back through this calculation many times you might notice that the number followed a pattern, a rule of thumb to calculate without tables and calculators, and that turns out to be 3/(n+1), the statistical rule of three. If you wanted greater certainty, you can use the rule of seven, which says that 7/(n+1) will give the 99% interval boundary. So in the case of your gumballs, you can be 99% sure the percentage of gumballs is less than 1/3.

The Rule(s) of Three, Including the Probability of Nothing

A Re-edit and Posting of a 2008 Blog

From 1827 Pike's Arithmetic

In my youth, back when dinosaurs roamed the earth, there was “the rule of three”… singular, one, and even then the name was often described as “archaic”. More modern books tended to develop “properties of proportions” or similar terms for the problems of proportionalities. Now there seem to be an abundance of them; including one for witches, and one about businesses. There is not space enough to talk about all of them so I will mention three, of course.
The first rule of three is as old as math, and shows up at least as early as the Hindu mathematician Brahmagupta, and in Fibonacci’s famous Liber Abaci(1202). It was once so common that it was introduced into common language. Abraham Lincoln is quoted in his biography as stating that he learned to "read, write, and cipher to the rule of 3."   So common that student's often wrote verse like the following, in their copy (practice) books.

Multiplication is vexation;

Division is as bad;

The Rule of Three doth puzzle me,

And Practice drives me mad

The most common and longest living form was the direct rule (although there was an inverse rule as well), in which case three numbers would be given and a fourth sought so that the ratio between the third and fourth would match the ratio between the first and second; a:b = c:d. Today students use the ideas in elementary school to complete fraction equivalences, “2/3 is the same as 10/?” Some of the ancient examples grew incredibly complicated.

I suppose the reason I chose to address three of the many “rules of three” is because of the rule of three from language and literature. Three just seems to be the right number for lots of things, there were Three Musketeers, Three Stooges, and Three Coins in the Fountain. It was Goldilocks and the Three Bears, and “bah bah black sheep” had “three bags full.” Comics in the newspaper usually have three panels and many jokes involve a three part ritual where the punch line is the third element, such as the t-shirt with “Great Cities of the World” on the top, and below, one after another, “Paris, Rome, Fargo”. The first two make the last funnier. In language the examples range from “Blood, sweat, and tears, to vidi, vidi, vici. If you don’t think there really is a mental tendency to have three terms, consider that in Churchill’s speech, he actually used four; “I say to the House as I said to ministers who have joined this government, I have nothing to offer but blood, toil, tears, and sweat. “

The final rule of three I would mention is from statistics, and is of more recent origin. It is also, I think, a really clever solution to what is a really difficult problem. Suppose something never happens; how can you assign a probability to it? It is not that it might not happen some day, just not so far. It is just such a problem the statistical rule of there was created to handle. Suppose you stopped at the same gum ball machine every day, but unlike the normal gumball machine, this one did not have a glass you could see into the gumballs inside. You buy a gum ball every day and get red ones, and green ones, but never a blue one. After a while you begin to wonder if they even put a blue one in the machine. So one day, after 20 days of getting all the other colors, over lunch you ask your local statistician (doesn’t everyone have lunch with a statistician?) how to figure out if there really is a blue one in there. He pauses, fork poised in mid-air, and informs you that you can be 95% sure (a common statistical benchmark) that the proportion of blue gum balls is no greater than 14.3%. He had mentally taken three, and divided by one more than the number of failed efforts, to get 3/21 or 1/7 as the upper limit of the possible fraction.
The idea is base on a simple extension of the binomial probability. If you knew that P % of the gum balls were blue, then you could calculate the probability that None showed up in 20 days. The probability would be (1-p)²⁰. Working back through this calculation many times you might notice that the number followed a pattern, a rule of thumb to calculate without tables and calculators, and that turns out to be 3/(n+1), the statistical rule of three. If you wanted greater certainty, you can use the rule of seven, which says that 7/(n+1) will give the 99% interval boundary. So in the case of your gumballs, you can be 99% sure the percentage of gumballs is less than 1/3.

The Rule(s) of Three and the Probability of Nothing

A Re-edit and Posting of a 2008 Blog

From 1827 Pike's Arithmetic

In my youth, back when dinosaurs roamed the earth, there was “the rule of three”… singular, one, and even then the name was often described as “archaic”. More modern books tended to develop “properties of proportions” or similar terms for the problems of proportionalities. Now there seem to be an abundance of them; including one for witches, and one about businesses. There is not space enough to talk about all of them so I will mention three, of course.
The first rule of three is as old as math, and shows up at least as early as the Hindu mathematician Brahmagupta, and in Fibonacci’s famous Liber Abaci(1202). It was once so common that it was introduced into common language. Abraham Lincoln is quoted in his biography as stating that he learned to "read, write, and cipher to the rule of 3."   So common that student's often wrote verse like the following, in their copy (practice) books.

Multiplication is vexation;

Division is as bad;

The Rule of Three doth puzzle me,

And Practice drives me mad

The most common and longest living form was the direct rule (although there was an inverse rule as well), in which case three numbers would be given and a fourth sought so that the ratio between the third and fourth would match the ratio between the first and second; a:b = c:d. Today students use the ideas in elementary school to complete fraction equivalences, “2/3 is the same as 10/?” Some of the ancient examples grew incredibly complicated.

I suppose the reason I chose to address three of the many “rules of three” is because of the rule of three from language and literature. Three just seems to be the right number for lots of things, there were Three Musketeers, Three Stooges, and Three Coins in the Fountain. It was Goldilocks and the Three Bears, and “bah bah black sheep” had “three bags full.” Comics in the newspaper usually have three panels and many jokes involve a three part ritual where the punch line is the third element, such as the t-shirt with “Great Cities of the World” on the top, and below, one after another, “Paris, Rome, Fargo”. The first two make the last funnier. In language the examples range from “Blood, sweat, and tears, to vidi, vidi, vici. If you don’t think there really is a mental tendency to have three terms, consider that in Churchill’s speech, he actually used four; “I say to the House as I said to ministers who have joined this government, I have nothing to offer but blood, toil, tears, and sweat. “

The final rule of three I would mention is from statistics, and is of more recent origin. It is also, I think, a really clever solution to what is a really difficult problem. Suppose something never happens; how can you assign a probability to it? It is not that it might not happen some day, just not so far. It is just such a problem the statistical rule of there was created to handle. Suppose you stopped at the same gum ball machine every day, but unlike the normal gumball machine, this one did not have a glass you could see into the gumballs inside. You buy a gum ball every day and get red ones, and green ones, but never a blue one. After a while you begin to wonder if they even put a blue one in the machine. So one day, after 20 days of getting all the other colors, over lunch you ask your local statistician (doesn’t everyone have lunch with a statistician?) how to figure out if there really is a blue one in there. He pauses, fork poised in mid-air, and informs you that you can be 95% sure (a common statistical benchmark) that the proportion of blue gum balls is no greater than 14.3%. He had mentally taken three, and divided by one more than the number of failed efforts, to get 3/21 or 1/7 as the upper limit of the possible fraction.
The idea is base on a simple extension of the binomial probability. If you knew that P % of the gum balls were blue, then you could calculate the probability that None showed up in 20 days. The probability would be (1-p)²⁰. Working back through this calculation many times you might notice that the number followed a pattern, a rule of thumb to calculate without tables and calculators, and that turns out to be 3/(n+1), the statistical rule of three. If you wanted greater certainty, you can use the rule of seven, which says that 7/(n+1) will give the 99% interval boundary. So in the case of your gumballs, you can be 99% sure the percentage of gumballs is less than 1/3.

The Rules of Three

In my youth, back when dinosaurs roamed the earth, there was “the rule of three”… singular, one, and even then the name was often described as “archaic”. More modern books tended to develop “properties of proportions” or similar terms for the problems of proportionalities. Now there seem to be an abundance of them; including one for witches, and one about businesses. There is not space enough to talk about all of them so I will mention three, of course.
The first rule of three is as old as math, and shows up at least as early as the Hindu mathematician Brahmagupta, and in Fibonacci’s famous Liber Abaci(1202). It was once so common that it was introduced into common language. Abraham Lincoln is quoted in his biography as stating that he learned to "read, write, and cipher to the rule of 3."
The most common and longest living form was the direct rule (although there was an inverse rule as well), in which case three numbers would be given and a fourth sought so that the ratio between the third and fourth would match the ratio between the first and second; a:b = c:d. Today students use the ideas in elementary school to complete fraction equivalences, “2/3 is the same as 10/?” Some of the ancient examples grew incredibly complicated.

I suppose the reason I chose to address three of the many “rules of three” is because of the rule of three from language and literature. Three just seems to be the right number for lots of things, there were Three Musketeers, Three Stooges, and Three Coins in the Fountain. It was Goldilocks and the Three Bears, and “bah bah black sheep” had “three bags full.” Comics in the newspaper usually have three panels and many jokes involve a three part ritual where the punch line is the third element, such as the t-shirt with “Great Cities of the World” on the top, and below, one after another, “Paris, Rome, Fargo”. The first two make the last funnier. In language the examples range from “Blood, sweat, and tears, to vidi, vidi, vici. If you don’t think there really is a mental tendency to have three terms, consider that in Churchill’s speech, he actually used four; “I say to the House as I said to ministers who have joined this government, I have nothing to offer but blood, toil, tears, and sweat. “
The final rule of three I would mention is from statistics, and is of more recent origin. It is also, I think, a really clever solution to what is a really difficult problem. Suppose something never happens; how can you assign a probability to it? It is not that it might not happen some day, just not so far. It is just such a problem the statistical rule of there was created to handle. Suppose you stopped at the same gum ball machine every day, but unlike the normal gumball machine, this one did not have a glass you could see into the gumballs inside. You buy a gum ball every day and get red ones, and green ones, but never a blue one. After a while you begin to wonder if they even put a blue one in the machine. So one day, after 20 days of getting all the other colors, over lunch you ask your local statistician (doesn’t everyone have lunch with a statistician?) how to figure out if there really is a blue one in there. He pauses, fork poised in mid-air, and informs you that you can be 95% sure (a common statistical benchmark) that the proportion of blue gum balls is no greater than 14.3%. He had mentally taken three, and divided by one more than the number of failed efforts, to get 3/21 or 1/7 as the upper limit of the possible fraction.
The idea is base on a simple extension of the binomial probability. If you knew that P % of the gum balls were blue, then you could calculate the probability that None showed up in 20 days. The probability would be (1-p)²⁰. Working back through this calculation many times you might notice that the number followed a pattern, a rule of thumb to calculate without tables and calculators, and that turns out to be 3/(n+1), the statistical rule of three. If you wanted greater certainty, you can use the rule of seven, which says that 7/(n+1) will give the 99% interval boundary. So in the case of your gumballs, you can be 99% sure the percentage of gumballs is less than 1/3.

The Rule(s) of Three and the Probability of Nothing, and of a single success

A Re-edit and Posting of a 2008 Blog with additional material (See, I do learn something over time.)

From 1827 Pike's Arithmetic

In my youth, back when dinosaurs roamed the earth, there was “the rule of three”… singular, one, and even then the name was often described as “archaic”. More modern books tended to develop “properties of proportions” or similar terms for the problems of proportionalities. Now there seem to be an abundance of them; including one for witches, and one about businesses. There is not space enough to talk about all of them so I will mention three, of course.
The first rule of three is as old as math, and shows up at least as early as the Hindu mathematician Brahmagupta, and in Fibonacci’s famous Liber Abaci(1202). It was once so common that it was introduced into common language. Abraham Lincoln is quoted in his biography as stating that he learned to "read, write, and cipher to the rule of 3."   So common that student's often wrote verse like the following, in their copy (practice) books.

Multiplication is vexation;

Division is as bad;

The Rule of Three doth puzzle me,

And Practice drives me mad

The most common and longest living form was the direct rule (although there was an inverse rule as well), in which case three numbers would be given and a fourth sought so that the ratio between the third and fourth would match the ratio between the first and second; a:b = c:d. Today students use the ideas in elementary school to complete fraction equivalences, “2/3 is the same as 10/?” Some of the ancient examples grew incredibly complicated.

I suppose the reason I chose to address three of the many “rules of three” is because of the rule of three from language and literature. Three just seems to be the right number for lots of things, there were Three Musketeers, Three Stooges, and Three Coins in the Fountain. It was Goldilocks and the Three Bears, and “bah bah black sheep” had “three bags full.” Comics in the newspaper usually have three panels and many jokes involve a three part ritual where the punch line is the third element, such as the t-shirt with “Great Cities of the World” on the top, and below, one after another, “Paris, Rome, Fargo”. The first two make the last funnier. In language the examples range from “Blood, sweat, and tears, to vidi, vidi, vici. If you don’t think there really is a mental tendency to have three terms, consider that in Churchill’s speech, he actually used four; “I say to the House as I said to ministers who have joined this government, I have nothing to offer but blood, toil, tears, and sweat. “

The final rule of three I would mention is from statistics, and is of more recent origin. It is also, I think, a really clever solution to what is a really difficult problem. Suppose something never happens; how can you assign a probability to it? It is not that it might not happen some day, just not so far. It is just such a problem the statistical rule of there was created to handle. Suppose you stopped at the same gum ball machine every day, but unlike the normal gumball machine, this one did not have a glass you could see into the gumballs inside. You buy a gum ball every day and get red ones, and green ones, but never a blue one. After a while you begin to wonder if they even put a blue one in the machine. So one day, after 20 days of getting all the other colors, over lunch you ask your local statistician (doesn’t everyone have lunch with a statistician?) how to figure out if there really is a blue one in there. He pauses, fork poised in mid-air, and informs you that you can be 95% sure (a common statistical benchmark) that the proportion of blue gum balls is no greater than 14.3%. He had mentally taken three, and divided by one more than the number of failed efforts, to get 3/21 or 1/7 as the upper limit of the possible fraction.
The idea is base on a simple extension of the binomial probability. If you knew that P % of the gum balls were blue, then you could calculate the probability that None showed up in 20 days. The probability would be (1-p)²⁰. Working back through this calculation many times you might notice that the number followed a pattern, a rule of thumb to calculate without tables and calculators, and that turns out to be 3/(n+1), the statistical rule of three. If you wanted greater certainty, you can use the rule of seven, which says that 7/(n+1) will give the 99% interval boundary. So in the case of your gumballs, you can be 99% sure the percentage of blue gumballs is less than 1/3.

But what if after a long string of failures, you have a success.  How does this change your confidence interval?   Thanks to a recent post from John D Cook I now can tell you that as well.  

So suppose you had worked your way as before with twenty failures to get the blue gumball, and then after the aforementioned lunch with a statistician,  you get a blue gumball on the 21st try.  Now what can you say about the expected percentage of blue gumballs.  

After the first success according to the Beta distribution would give a 95% confidence interval of appx. [.1/n, 4.7/n]  .  For our imagined 21 tries, this would be about [.0047, .224]  So our confidence interval has opened up considerably.  

It appears, if I understand correctly, that the blue gumball could have occurred anytime among the first 21 tries and thus would still be the CI.  So if we went another nine tries without success, we would adjust our CI to {.1/30, 4.7/30] ... [ .00333, .157], back much closer to our expectations before we ever had a success.

Comparing this interval to the binomial confidence interval you learned in high school math, p +/- 2 sqrt(p*(1-p)/n).  The customary warning on the normal expectation is beware of p being too high or too low.  Using one success in 30 tries we get a 95% CI of [-.03, .099]... perhaps the negative lower bound is a sign that we have strayed to close to zero with our p-hat.  A nice topic to spring on your AP stats teacher when you get to confidence intervals, but please be kind. 

The Rule(s) of Three and the Probability of Nothing, and of a single success

 

A Re-edit and Posting of a 2008 Blog with additional material (See, I do learn something over time.)

From 1827 Pike's Arithmetic

In my youth, back when dinosaurs roamed the earth, there was “the rule of three”… singular, one, and even then the name was often described as “archaic”. More modern books tended to develop “properties of proportions” or similar terms for the problems of proportionalities. Now there seem to be an abundance of them; including one for witches, and one about businesses. There is not space enough to talk about all of them so I will mention three, of course.
The first rule of three is as old as math, and shows up at least as early as the Hindu mathematician Brahmagupta, and in Fibonacci’s famous Liber Abaci(1202). It was once so common that it was introduced into common language. Abraham Lincoln is quoted in his biography as stating that he learned to "read, write, and cipher to the rule of 3."   So common that student's often wrote verse like the following, in their copy (practice) books.

Multiplication is vexation;

Division is as bad;

The Rule of Three doth puzzle me,

And Practice drives me mad

The most common and longest living form was the direct rule (although there was an inverse rule as well), in which case three numbers would be given and a fourth sought so that the ratio between the third and fourth would match the ratio between the first and second; a:b = c:d. Today students use the ideas in elementary school to complete fraction equivalences, “2/3 is the same as 10/?” Some of the ancient examples grew incredibly complicated.

I suppose the reason I chose to address three of the many “rules of three” is because of the rule of three from language and literature. Three just seems to be the right number for lots of things, there were Three Musketeers, Three Stooges, and Three Coins in the Fountain. It was Goldilocks and the Three Bears, and “bah bah black sheep” had “three bags full.” Comics in the newspaper usually have three panels and many jokes involve a three part ritual where the punch line is the third element, such as the t-shirt with “Great Cities of the World” on the top, and below, one after another, “Paris, Rome, Fargo”. The first two make the last funnier. In language the examples range from “Blood, sweat, and tears, to vidi, vidi, vici. If you don’t think there really is a mental tendency to have three terms, consider that in Churchill’s speech, he actually used four; “I say to the House as I said to ministers who have joined this government, I have nothing to offer but blood, toil, tears, and sweat. “

The final rule of three I would mention is from statistics, and is of more recent origin. It is also, I think, a really clever solution to what is a really difficult problem. Suppose something never happens; how can you assign a probability to it? It is not that it might not happen some day, just not so far. It is just such a problem the statistical rule of there was created to handle. Suppose you stopped at the same gum ball machine every day, but unlike the normal gumball machine, this one did not have a glass you could see into the gumballs inside. You buy a gum ball every day and get red ones, and green ones, but never a blue one. After a while you begin to wonder if they even put a blue one in the machine. So one day, after 20 days of getting all the other colors, over lunch you ask your local statistician (doesn’t everyone have lunch with a statistician?) how to figure out if there really is a blue one in there. He pauses, fork poised in mid-air, and informs you that you can be 95% sure (a common statistical benchmark) that the proportion of blue gum balls is no greater than 14.3%. He had mentally taken three, and divided by one more than the number of failed efforts, to get 3/21 or 1/7 as the upper limit of the possible fraction.
The idea is base on a simple extension of the binomial probability. If you knew that P % of the gum balls were blue, then you could calculate the probability that None showed up in 20 days. The probability would be (1-p)²⁰. Working back through this calculation many times you might notice that the number followed a pattern, a rule of thumb to calculate without tables and calculators, and that turns out to be 3/(n+1), the statistical rule of three. If you wanted greater certainty, you can use the rule of seven, which says that 7/(n+1) will give the 99% interval boundary. So in the case of your gumballs, you can be 99% sure the percentage of blue gumballs is less than 1/3.

But what if after a long string of failures, you have a success.  How does this change your confidence interval?   Thanks to a recent post from John D Cook I now can tell you that as well.  

So suppose you had worked your way as before with twenty failures to get the blue gumball, and then after the aforementioned lunch with a statistician,  you get a blue gumball on the 21st try.  Now what can you say about the expected percentage of blue gumballs.  

After the first success according to the Beta distribution would give a 95% confidence interval of appx. [.1/n, 4.7/n]  .  For our imagined 21 tries, this would be about [.0047, .224]  So our confidence interval has opened up considerably.  

It appears, if I understand correctly, that the blue gumball could have occurred anytime among the first 21 tries and thus would still be the CI.  So if we went another nine tries without success, we would adjust our CI to {.1/30, 4.7/30] ... [ .00333, .157], back much closer to our expectations before we ever had a success.

Comparing this interval to the binomial confidence interval you learned in high school math, p +/- 2 sqrt(p*(1-p)/n).  The customary warning on the normal expectation is beware of p being too high or too low.  Using one success in 30 tries we get a 95% CI of [-.03, .099]... perhaps the negative lower bound is a sign that we have strayed to close to zero with our p-hat.  A nice topic to spring on your AP stats teacher when you get to confidence intervals, but please be kind. 

The Rule(s) of Three and the Probability of Nothing

A Re-edit and Posting of a 2008 Blog

From 1827 Pike's Arithmetic

In my youth, back when dinosaurs roamed the earth, there was “the rule of three”… singular, one, and even then the name was often described as “archaic”. More modern books tended to develop “properties of proportions” or similar terms for the problems of proportionalities. Now there seem to be an abundance of them; including one for witches, and one about businesses. There is not space enough to talk about all of them so I will mention three, of course.
The first rule of three is as old as math, and shows up at least as early as the Hindu mathematician Brahmagupta, and in Fibonacci’s famous Liber Abaci(1202). It was once so common that it was introduced into common language. Abraham Lincoln is quoted in his biography as stating that he learned to "read, write, and cipher to the rule of 3."   So common that student's often wrote verse like the following, in their copy (practice) books.

Multiplication is vexation;

Division is as bad;

The Rule of Three doth puzzle me,

And Practice drives me mad

The most common and longest living form was the direct rule (although there was an inverse rule as well), in which case three numbers would be given and a fourth sought so that the ratio between the third and fourth would match the ratio between the first and second; a:b = c:d. Today students use the ideas in elementary school to complete fraction equivalences, “2/3 is the same as 10/?” Some of the ancient examples grew incredibly complicated.

I suppose the reason I chose to address three of the many “rules of three” is because of the rule of three from language and literature. Three just seems to be the right number for lots of things, there were Three Musketeers, Three Stooges, and Three Coins in the Fountain. It was Goldilocks and the Three Bears, and “bah bah black sheep” had “three bags full.” Comics in the newspaper usually have three panels and many jokes involve a three part ritual where the punch line is the third element, such as the t-shirt with “Great Cities of the World” on the top, and below, one after another, “Paris, Rome, Fargo”. The first two make the last funnier. In language the examples range from “Blood, sweat, and tears, to veni, vidi, vici. If you don’t think there really is a mental tendency to have three terms, consider that in Churchill’s speech, he actually used four; “I say to the House as I said to ministers who have joined this government, I have nothing to offer but blood, toil, tears, and sweat. “  But who cares what he said, you will hear the phrase almost always as "Blood, sweat, and tears, and it was common usage before and (however long they last) after the band.

The final rule of three I would mention is from statistics, and is of more recent origin. It is also, I think, a really clever solution to what is a really difficult problem. Suppose something never happens; how can you assign a probability to it? It is not that it might not happen some day, just not so far. It is just such a problem the statistical rule of there was created to handle. Suppose you stopped at the same gum ball machine every day, but unlike the normal gumball machine, this one did not have a glass you could see into the gumballs inside. You buy a gum ball every day and get red ones, and green ones, but never a blue one. After a while you begin to wonder if they even put a blue one in the machine. So one day, after 20 days of getting all the other colors, over lunch you ask your local statistician (doesn’t everyone have lunch with a statistician?) how to figure out if there really is a blue one in there. He pauses, fork poised in mid-air, and informs you that you can be 95% sure (a common statistical benchmark) that the proportion of blue gum balls is no greater than 14.3%. He had mentally taken three, and divided by one more than the number of failed efforts, to get 3/21 or 1/7 as the upper limit of the possible fraction.
The idea is base on a simple extension of the binomial probability. If you knew that P % of the gum balls were blue, then you could calculate the probability that None showed up in 20 days. The probability would be (1-p)²⁰. Working back through this calculation many times you might notice that the number followed a pattern, a rule of thumb to calculate without tables and calculators, and that turns out to be 3/(n+1), the statistical rule of three, giving a probability for the Blue gum balls as 0< P < .143.  If you wanted greater certainty, you can use the rule of seven, which says that 7/(n+1) will give the 99% interval boundary. So in the case of your gumballs, you can be 99% sure the percentage of blue gumballs is less than 1/3. (Of course this problem assumes a population that constantly replaces the gum ball removed so that the probabilities remain constant.

But what if after a long string of failures, you have a success.  How does this change your confidence interval?   Thanks to a recent post from John D Cook I now can tell you that as well.  

So suppose you had worked your way as before with twenty failures to get the blue gumball, and then after the aforementioned lunch with a statistician,  you get a blue gumball on the 21st try.  Now what can you say about the expected percentage of blue gumballs.  

After the first success according to the Beta distribution would give a 95% confidence interval of appx. [.1/n, 4.7/n]  .  For our imagined 21 tries, this would be about [.0047, .224]  So our confidence interval has opened up considerably.  

It appears, if I understand correctly, that the blue gumball could have occurred anytime among the first 21 tries and thus would still be the CI.  So if we went another nine tries without success, we would adjust our CI to {.1/30, 4.7/30] ... [ .00333, .157], back much closer to our expectations before we ever had a success.

Comparing this interval to the binomial confidence interval you learned in high school math, p +/- 2 sqrt(p*(1-p)/n).  The customary warning on the normal expectation is beware of p being too high or too low.  Using one success in 30 tries we get a 95% CI of [-.03, .099]... perhaps the negative lower bound is a sign that we have strayed to close to zero with our p-hat.  A nice topic to spring on your AP stats teacher when you get to confidence intervals, but please be kind. 

 

The Rule(s) of Three and the Probability of Nothing

A Re-edit and Posting of a 2008 Blog

From 1827 Pike's Arithmetic

In my youth, back when dinosaurs roamed the earth, there was “the rule of three”… singular, one, and even then the name was often described as “archaic”. More modern books tended to develop “properties of proportions” or similar terms for the problems of proportionalities. Now there seem to be an abundance of them; including one for witches, and one about businesses. There is not space enough to talk about all of them so I will mention three, of course.
The first rule of three is as old as math, and shows up at least as early as the Hindu mathematician Brahmagupta, and in Fibonacci’s famous Liber Abaci(1202). It was once so common that it was introduced into common language. Abraham Lincoln is quoted in his biography as stating that he learned to "read, write, and cipher to the rule of 3."   So common that student's often wrote verse like the following, in their copy (practice) books.

Multiplication is vexation;

Division is as bad;

The Rule of Three doth puzzle me,

And Practice drives me mad

The most common and longest living form was the direct rule (although there was an inverse rule as well), in which case three numbers would be given and a fourth sought so that the ratio between the third and fourth would match the ratio between the first and second; a:b = c:d. Today students use the ideas in elementary school to complete fraction equivalences, “2/3 is the same as 10/?” Some of the ancient examples grew incredibly complicated.

I suppose the reason I chose to address three of the many “rules of three” is because of the rule of three from language and literature. Three just seems to be the right number for lots of things, there were Three Musketeers, Three Stooges, and Three Coins in the Fountain. It was Goldilocks and the Three Bears, and “bah bah black sheep” had “three bags full.” Comics in the newspaper usually have three panels and many jokes involve a three part ritual where the punch line is the third element, such as the t-shirt with “Great Cities of the World” on the top, and below, one after another, “Paris, Rome, Fargo”. The first two make the last funnier. In language the examples range from “Blood, sweat, and tears, to veni, vidi, vici. If you don’t think there really is a mental tendency to have three terms, consider that in Churchill’s speech, he actually used four; “I say to the House as I said to ministers who have joined this government, I have nothing to offer but blood, toil, tears, and sweat. “  But who cares what he said, you will hear the phrase almost always as "Blood, sweat, and tears, and it was common usage before and (however long they last) after the band.

The final rule of three I would mention is from statistics, and is of more recent origin. It is also, I think, a really clever solution to what is a really difficult problem. Suppose something never happens; how can you assign a probability to it? It is not that it might not happen some day, just not so far. It is just such a problem the statistical rule of there was created to handle. Suppose you stopped at the same gum ball machine every day, but unlike the normal gumball machine, this one did not have a glass you could see into the gumballs inside. You buy a gum ball every day and get red ones, and green ones, but never a blue one. After a while you begin to wonder if they even put a blue one in the machine. So one day, after 20 days of getting all the other colors, over lunch you ask your local statistician (doesn’t everyone have lunch with a statistician?) how to figure out if there really is a blue one in there. He pauses, fork poised in mid-air, and informs you that you can be 95% sure (a common statistical benchmark) that the proportion of blue gum balls is no greater than 14.3%. He had mentally taken three, and divided by one more than the number of failed efforts, to get 3/21 or 1/7 as the upper limit of the possible fraction.
The idea is base on a simple extension of the binomial probability. If you knew that P % of the gum balls were blue, then you could calculate the probability that None showed up in 20 days. The probability would be (1-p)²⁰. Working back through this calculation many times you might notice that the number followed a pattern, a rule of thumb to calculate without tables and calculators, and that turns out to be 3/(n+1), the statistical rule of three, giving a probability for the Blue gum balls as 0< P < .143.  If you wanted greater certainty, you can use the rule of seven, which says that 7/(n+1) will give the 99% interval boundary. So in the case of your gumballs, you can be 99% sure the percentage of blue gumballs is less than 1/3. (Of course this problem assumes a population that constantly replaces the gum ball removed so that the probabilities remain constant.

But what if after a long string of failures, you have a success.  How does this change your confidence interval?   Thanks to a recent post from John D Cook I now can tell you that as well.  

So suppose you had worked your way as before with twenty failures to get the blue gumball, and then after the aforementioned lunch with a statistician,  you get a blue gumball on the 21st try.  Now what can you say about the expected percentage of blue gumballs.  

After the first success according to the Beta distribution would give a 95% confidence interval of appx. [.1/n, 4.7/n]  .  For our imagined 21 tries, this would be about [.0047, .224]  So our confidence interval has opened up considerably.  

It appears, if I understand correctly, that the blue gumball could have occurred anytime among the first 21 tries and thus would still be the CI.  So if we went another nine tries without success, we would adjust our CI to {.1/30, 4.7/30] ... [ .00333, .157], back much closer to our expectations before we ever had a success.

Comparing this interval to the binomial confidence interval you learned in high school math, p +/- 2 sqrt(p*(1-p)/n).  The customary warning on the normal expectation is beware of p being too high or too low.  Using one success in 30 tries we get a 95% CI of [-.03, .099]... perhaps the negative lower bound is a sign that we have strayed to close to zero with our p-hat.  A nice topic to spring on your AP stats teacher when you get to confidence intervals, but please be kind. 

 

The Rule(s) of Three and the Probability of Nothing

  

A Re-edit and Posting of a 2008 Blog

From 1827 Pike's Arithmetic

In my youth, back when dinosaurs roamed the earth, there was “the rule of three”… singular, one, and even then the name was often described as “archaic”. More modern books tended to develop “properties of proportions” or similar terms for the problems of proportionalities. Now there seem to be an abundance of them; including one for witches, and one about businesses. There is not space enough to talk about all of them so I will mention three, of course.
The first rule of three is as old as math, and shows up at least as early as the Hindu mathematician Brahmagupta, and in Fibonacci’s famous Liber Abaci(1202). It was once so common that it was introduced into common language. Abraham Lincoln is quoted in his biography as stating that he learned to "read, write, and cipher to the rule of 3."   So common that student's often wrote verse like the following, in their copy (practice) books.

Multiplication is vexation;

Division is as bad;

The Rule of Three doth puzzle me,

And Practice drives me mad

The most common and longest living form was the direct rule (although there was an inverse rule as well), in which case three numbers would be given and a fourth sought so that the ratio between the third and fourth would match the ratio between the first and second; a:b = c:d. Today students use the ideas in elementary school to complete fraction equivalences, “2/3 is the same as 10/?” Some of the ancient examples grew incredibly complicated.

I suppose the reason I chose to address three of the many “rules of three” is because of the rule of three from language and literature. Three just seems to be the right number for lots of things, there were Three Musketeers, Three Stooges, and Three Coins in the Fountain. It was Goldilocks and the Three Bears, and “bah bah black sheep” had “three bags full.” Comics in the newspaper usually have three panels and many jokes involve a three part ritual where the punch line is the third element, such as the t-shirt with “Great Cities of the World” on the top, and below, one after another, “Paris, Rome, Fargo”. The first two make the last funnier. In language the examples range from “Blood, sweat, and tears, to veni, vidi, vici. If you don’t think there really is a mental tendency to have three terms, consider that in Churchill’s speech, he actually used four; “I say to the House as I said to ministers who have joined this government, I have nothing to offer but blood, toil, tears, and sweat. “  But who cares what he said, you will hear the phrase almost always as "Blood, sweat, and tears, and it was common usage before and (however long they last) after the band.

The final rule of three I would mention is from statistics, and is of more recent origin. It is also, I think, a really clever solution to what is a really difficult problem. Suppose something never happens; how can you assign a probability to it? It is not that it might not happen some day, just not so far. It is just such a problem the statistical rule of there was created to handle. Suppose you stopped at the same gum ball machine every day, but unlike the normal gumball machine, this one did not have a glass you could see into the gumballs inside. You buy a gum ball every day and get red ones, and green ones, but never a blue one. After a while you begin to wonder if they even put a blue one in the machine. So one day, after 20 days of getting all the other colors, over lunch you ask your local statistician (doesn’t everyone have lunch with a statistician?) how to figure out if there really is a blue one in there. He pauses, fork poised in mid-air, and informs you that you can be 95% sure (a common statistical benchmark) that the proportion of blue gum balls is no greater than 14.3%. He had mentally taken three, and divided by one more than the number of failed efforts, to get 3/21 or 1/7 as the upper limit of the possible fraction.
The idea is base on a simple extension of the binomial probability. If you knew that P % of the gum balls were blue, then you could calculate the probability that None showed up in 20 days. The probability would be (1-p)²⁰. Working back through this calculation many times you might notice that the number followed a pattern, a rule of thumb to calculate without tables and calculators, and that turns out to be 3/(n+1), the statistical rule of three, giving a probability for the Blue gum balls as 0< P < .143.  If you wanted greater certainty, you can use the rule of seven, which says that 7/(n+1) will give the 99% interval boundary. So in the case of your gumballs, you can be 99% sure the percentage of blue gumballs is less than 1/3. (Of course this problem assumes a population that constantly replaces the gum ball removed so that the probabilities remain constant.

But what if after a long string of failures, you have a success.  How does this change your confidence interval?   Thanks to a recent post from John D Cook I now can tell you that as well.  

So suppose you had worked your way as before with twenty failures to get the blue gumball, and then after the aforementioned lunch with a statistician,  you get a blue gumball on the 21st try.  Now what can you say about the expected percentage of blue gumballs.  

After the first success according to the Beta distribution would give a 95% confidence interval of appx. [.1/n, 4.7/n]  .  For our imagined 21 tries, this would be about [.0047, .224]  So our confidence interval has opened up considerably.  

It appears, if I understand correctly, that the blue gumball could have occurred anytime among the first 21 tries and thus would still be the CI.  So if we went another nine tries without success, we would adjust our CI to {.1/30, 4.7/30] ... [ .00333, .157], back much closer to our expectations before we ever had a success.

Comparing this interval to the binomial confidence interval you learned in high school math, p +/- 2 sqrt(p*(1-p)/n).  The customary warning on the normal expectation is beware of p being too high or too low.  Using one success in 30 tries we get a 95% CI of [-.03, .099]... perhaps the negative lower bound is a sign that we have strayed to close to zero with our p-hat.  A nice topic to spring on your AP stats teacher when you get to confidence intervals, but please be kind. 

 

The Rule(s) of Three and the Probability of Nothing

 

A Re-edit and Posting of a 2008 Blog

From 1827 Pike's Arithmetic

In my youth, back when dinosaurs roamed the earth, there was “the rule of three”… singular, one, and even then the name was often described as “archaic”. More modern books tended to develop “properties of proportions” or similar terms for the problems of proportionalities. Now there seem to be an abundance of them; including one for witches, and one about businesses. There is not space enough to talk about all of them so I will mention three, of course.
The first rule of three is as old as math, and shows up at least as early as the Hindu mathematician Brahmagupta, and in Fibonacci’s famous Liber Abaci(1202). It was once so common that it was introduced into common language. Abraham Lincoln is quoted in his biography as stating that he learned to "read, write, and cipher to the rule of 3."   So common that student's often wrote verse like the following, in their copy (practice) books.

Multiplication is vexation;

Division is as bad;

The Rule of Three doth puzzle me,

And Practice drives me mad

The most common and longest living form was the direct rule (although there was an inverse rule as well), in which case three numbers would be given and a fourth sought so that the ratio between the third and fourth would match the ratio between the first and second; a:b = c:d. Today students use the ideas in elementary school to complete fraction equivalences, “2/3 is the same as 10/?” Some of the ancient examples grew incredibly complicated.

I suppose the reason I chose to address three of the many “rules of three” is because of the rule of three from language and literature. Three just seems to be the right number for lots of things, there were Three Musketeers, Three Stooges, and Three Coins in the Fountain. It was Goldilocks and the Three Bears, and “bah bah black sheep” had “three bags full.” Comics in the newspaper usually have three panels and many jokes involve a three part ritual where the punch line is the third element, such as the t-shirt with “Great Cities of the World” on the top, and below, one after another, “Paris, Rome, Fargo”. The first two make the last funnier. In language the examples range from “Blood, sweat, and tears, to veni, vidi, vici. If you don’t think there really is a mental tendency to have three terms, consider that in Churchill’s speech, he actually used four; “I say to the House as I said to ministers who have joined this government, I have nothing to offer but blood, toil, tears, and sweat. “  But who cares what he said, you will hear the phrase almost always as "Blood, sweat, and tears, and it was common usage before and (however long they last) after the band.

The final rule of three I would mention is from statistics, and is of more recent origin. It is also, I think, a really clever solution to what is a really difficult problem. Suppose something never happens; how can you assign a probability to it? It is not that it might not happen some day, just not so far. It is just such a problem the statistical rule of there was created to handle. Suppose you stopped at the same gum ball machine every day, but unlike the normal gumball machine, this one did not have a glass you could see into the gumballs inside. You buy a gum ball every day and get red ones, and green ones, but never a blue one. After a while you begin to wonder if they even put a blue one in the machine. So one day, after 20 days of getting all the other colors, over lunch you ask your local statistician (doesn’t everyone have lunch with a statistician?) how to figure out if there really is a blue one in there. He pauses, fork poised in mid-air, and informs you that you can be 95% sure (a common statistical benchmark) that the proportion of blue gum balls is no greater than 14.3%. He had mentally taken three, and divided by one more than the number of failed efforts, to get 3/21 or 1/7 as the upper limit of the possible fraction.
The idea is base on a simple extension of the binomial probability. If you knew that P % of the gum balls were blue, then you could calculate the probability that None showed up in 20 days. The probability would be (1-p)²⁰. Working back through this calculation many times you might notice that the number followed a pattern, a rule of thumb to calculate without tables and calculators, and that turns out to be 3/(n+1), the statistical rule of three, giving a probability for the Blue gum balls as 0< P < .143.  If you wanted greater certainty, you can use the rule of seven, which says that 7/(n+1) will give the 99% interval boundary. So in the case of your gumballs, you can be 99% sure the percentage of blue gumballs is less than 1/3. (Of course this problem assumes a population that constantly replaces the gum ball removed so that the probabilities remain constant.

But what if after a long string of failures, you have a success.  How does this change your confidence interval?   Thanks to a recent post from John D Cook I now can tell you that as well.  

So suppose you had worked your way as before with twenty failures to get the blue gumball, and then after the aforementioned lunch with a statistician,  you get a blue gumball on the 21st try.  Now what can you say about the expected percentage of blue gumballs.  

After the first success according to the Beta distribution would give a 95% confidence interval of appx. [.1/n, 4.7/n]  .  For our imagined 21 tries, this would be about [.0047, .224]  So our confidence interval has opened up considerably.  

It appears, if I understand correctly, that the blue gumball could have occurred anytime among the first 21 tries and thus would still be the CI.  So if we went another nine tries without success, we would adjust our CI to {.1/30, 4.7/30] ... [ .00333, .157], back much closer to our expectations before we ever had a success.

Comparing this interval to the binomial confidence interval you learned in high school math, p +/- 2 sqrt(p*(1-p)/n).  The customary warning on the normal expectation is beware of p being too high or too low.  Using one success in 30 tries we get a 95% CI of [-.03, .099]... perhaps the negative lower bound is a sign that we have strayed to close to zero with our p-hat.  A nice topic to spring on your AP stats teacher when you get to confidence intervals, but please be kind. 

 

"You Ain't Seen Nothing Yet", But You Might Soon

Apologies to Bachman Turner Overdrive...
In my recent blog about the solution to a variation of the birthday problem I promised to expand on the idea that if you haven't witnessed an occurrence of an event in a large (large enough) number of multiple trials, then we can say with confidence (95% statistically) that the probability of success will be no greater than 3/n. This is often called the "statistical rule of three."

If a binomial event has a probability of p, then the probability that after n trials there are no successes will be (1-p)ⁿ, you have to have a non-success every trial. And if you want that probability to be less than .05, then we have an exponential equation to solve, We take the natural log of both sides to make the problem simpler to solve,
Now if you have been wondering about where the three gets into it, this is it... the ln of .05 is very nearly -3. This gives us . Now we have to do a little calculus approximation. For very small values of p, ln(1-p) is very close to -p. Which means our equation is almost ..and dividing we can arrive at p=3/n.
So if we collect give 30 people an injection and none of them can experience a side effect, the rule of three suggests that we can be pretty (95%) sure that the true probability of the side effect is no greater than 1/10.

It seems to me that this approach is looking at the problem a little wrong. Let me try to explain. Let's take a very simple problem. Suppose we have a machine that spits out two ping-pong balls every time you press a button. The machine has five settings inside that control the probability that the ping-pong balls are white (otherwise they are some other color). The probabilities are equally spaced, 0, 1/4, 1/2, 3/4, and 1.
We want to know how how frequently we might get no white balls when we push the button. We make a table of the possible settings, and the probability that both balls are non-white.



Since the machine settings are random each of them has a probability of 1/5. To find the total probability of getting no whites, we multiply 1/5 times each probability and add them up. (1/5)(1)+(1/5)(9/16)+(1/5)(1/4)+(1/5)(1/16)+(1/5)(0)= 3/8. We would expect that, on average, 3/8 of the times we run the machine, we get two non-white balls.

But there is a different question that can be asked: "if someone got two non-white balls, what is the probability that the machine was set at some particular value (say p=1/2). This is the "conditional probability" that seems to make students crazy. But in this situation it look pretty clear. Of 80 people who ran the machine, 30 of them got two non-white balls. If we look just at this 30 people, we would see that only 1 of them had occurred when the machine was set to p=3/4, 4 when it was set to p=1/2, and the other 25 occurred when the machine was set to p=1/4 or p=0.

Notice that we can say two very similar sounding things that are quite different.
1) The probability you get two non-white balls IF the machine probability of white is set to p=1/2 or greater is less than 1/4. (I think this is the approach the rule of three uses)
2) The probability the machine is set to p(white)=1/2 or higher IF you get two white balls is 5/30 or 1/6. (I think this is the way the confidence interval should be calculated.)

Now if we extend the number of options of the machine probability of white to any real number in the interval 0 to 1, then we can answer the same questions using integration to find the area under the curve (1-p)² to give us the probability of getting two non-white balls. It turns out to be 1/3.

Now we can answer the same two questions as above
1) The probability you get two non-white balls IF the machine probability of white is set to p=1/2 or greater is less than 1/4. (This is exactly the same as before, we are just asking what is the function value when the input is 1/2)
2) The probability the machine is set to p(white)=1/2 or higher IF you get two white balls is the ratio of the shaded area to the right of x=1/2 to the total shaded area. It turns out that in this case (two failures) we get about .043/(1/3) or about 12%.

So what if we got three or four or more failures. The graph starts to look more and more skewed as the probability of failure clumps to the left of the curve. Here is the graph of (1-p)⁵ in red, and(1-p)¹⁰ in blue .


Our rule of three says that if we got ten consecutive failures, we can be pretty sure that the prob of success is probability greater than 3/10. I turns out that for p=3/10, the probability of getting ten failures in a row is about 2.8%, and even lower for any probability greater than 3/10. This suggests that even for relatively small values of n, the limit of 3/n is a conservative approach to bounding the probability of an occurrence. But is it the best estimate of the probability given that we have ten failures. Given that we have had ten failures, the probability that the success probability was 3/10 or higher is the ratio of the area under (1-p)¹⁰ between 3/10 and 1 (about .00179) over the area between 0 and 1 (that's 1/11 or .0909090..). The overall ratio is a little less than 2%.

The conclusion for me? The rule of three is a very over-cautious estimate, but well worth the effort when you consider the ease of computation.

Given Two Points????

  Each year in the spring my pre-calc kids come to the four brief sections in our text that deal with parametric equations and vectors. (I think of this chapter as the catch-all chapter, anything we might have missed that is in the California or Texas Standards). And then I throw in about three more weeks of work about vectors that I created because I think it is a) Each year in the spring my pre-calc kids come to the four brief sections in our text that deal with parametric equations and vectors. (I think of this chapter as the catch-all chapter, anything we might have missed that is in the California or Texas Standards). And then I throw in about three more weeks of work about vectors that I created because I think it is a)beautiful and b) really important. And I share with my students that it seems incredible to me that they, the best and brightest mathematics students in our school, are nearing the end of their high school education (many are seniors) and they can't do one of the most simple acts of coordinate geometry; that is, "Given two points, write the equation of a line containing the two points" (How does the standard for your school read?). 

 They look at me in wonder, confusion, and perhaps some doubt of my sanity. After all, we have done that thousands of times. I continue to bemoan their lack of ability until finally someone will challenge me...."But, Mr. B, we CAN do that. We do it all the time." Ok, We'll see, and I turn and write two points on the board... such as (3,1,2) and (2,4,3). They are so sure of their ability that they already have their pencils to paper when they realize they have no idea how to begin. I let them talk, explore, suggest ideas, and I wait, and I wait.... I have never had a student come up with an equation. Some will suggest it must be something like z=ax+by+c or something..... but NOT ONE ever hit upon a correct equation of the line in question.... Then we talk... Not about how to, that will come later, they will discover it on their own as a natural generalization, but about why not.. I admit to them that most of the students who graduate from high school (and in fact, many of the math teachers they have studied under) can not do this simple act of writing the equation of a line in the three space dimension that they live in. And I tell them that in the following weeks they will learn to do some of the simple geometry they know in the dimension they live in.

 I walk them through a simple vector approach to lines in the coordinate plane. We take y= 3x-1 and rewrite it as (x,y)= (0,-1) + t(1,3). Within minutes every kid in the class can write the equation of a line given two points in this vector form although a few struggle with the seeming reversal of order of the "slope"(in truth, several still mess up regularly when they try to use slope intercept, yet they seem reluctant to adopt the seemingly easier point-slope form). They are quickly taking two points and talking about "point vectors" and "slope vectors" as if they had used them forever. And each year it startles me anew that after a half-hour of an alternate approach, every student will intuitively generalize the method to produce a three-space equation of a line without any help...and then with a little faltering over the "fourth" variable, they can do the same thing in the barely imaginable four-space. Later we will write the equations of planes in space given three points and do some simple analytic geometry in three space. Many of them struggle with the idea of projections of lines and minor details, but I at least feel like I have made a small step to preparing them to function mathematically in the three-space they live in. And if the string-theory guys are right, and we really have a ten-dimensional universe.... no big deal, they can extend vectors to any dimension. But I wonder each year... why are we not introducing this more at an early age (alg I?). I will talk later about some of the advantages I see, and maybe you can tell me what I missing that would make it a bad idea. (Actually in the last few years I've been working a unit in parametric equations and vectors into my Alg II classes with lots of Bowditch curves (Lissajous?) and three D lines and planes and three D geometry ideas.)

I got this comment from one friend... 

Dave L. Renfro said...

Here's something marginally related to your topic. In 4 dimensional space it is possible for two planes to have exactly one point in common. A very simple example is the xy-plane, namely all points of the form (x,y,0,0) where x and y vary over the real numbers, and the zw-plane, namely all points of the form (0,0,z,w) where z and w vary over the real numbers. It is easy to see that these two planes have only the origin (0,0,0,0) in common, since x = y = z = w = 0 is the only solution to (x,y,0,0) = (0,0,z,w).

For what it's worth, I first learned about this possibility in a "side-bar diagram" in the following calculus book, which I was trying to work through during the 1974-75 school year (I managed to cover about the first 1/4 of the book), when I was in the 10th grade. Incidentally, the first college math class I took (an ODE course, about a year later) was from one of the authors (Embry) of this text.

Embry/Schell/Thomas, "Calculus and Linear Algebra. An Integrated Approach" (1972)

At this point you might try exploring other possible intersections between two four spaces, intersecting at a single point other than (0,0,0,0) , or a in a line, or a 3-plane.  Send me your best ideas, especially for sharing this with students.

This was all back in about 2009 and I retired two tears later and never followed up on this idea with a class, but I have some thoughts and will share them by adding to this post as time goes on, so check back...and share your ideas> I will pass them along as I go.  

I did come up with one more idea I wish I had thought of while I was teaching.  Somewhere in their HS math education most students who stay at it will learn to solve systems of equations and understand how to use matrices to find the point where three planes intersect IF they intersect in a single point.  (Thanks Dave!)

Planes in Space- part One

Take your basic bright kid in alg II or pre-calc, or often in calculus, and ask, "What is the intersection of two lines?"... They say, "A point."... good answer.

"Can you write the equation of a line?" Again they are on target.

"If I give you the equation of two lines on the plane; can you find their point of intersection?" The good ones can, and know they can.

Now ask the same bright kids, "What is the intersection of two planes in space?". They answer correctly again, "A line."

So far everything is great, but now we ask them to write the equation of a plane.... uhhh... gee..... and at this point, when asked about one of the fundamental structures of plane geometry, their analytic geometry skills are exhausted.

My experience was that after solving systems of three equations in three unknowns, they remembered how to do it (mostly) but didn't know what it was they had done???? 

but thankfully after a brief review a light goes on, and they understand.  

 Still, a very few may actually be able to produce x+y+z=1 or some other for the equation of a plane. Now we ask about the intersection of two planes, and almost none of them can do it. The scary part, is that very (very) few of the teachers of alg II and above that I have questioned about this could provide an answer either .

I would begin by recalling that an equation in three variables, such as 2x+3y+z=6 can represent a plane in space. When students had three such equations that intersected in a unique point, they found the solution by one of several methods. Most students learn to solve such equations by the methods called elimination and substitution at the very least. Others may have also been introduced to Cramer’s rule for solving systems with determinants and perhaps two methods using matrices.

The most commonly taught matrix method is to write a matrix equation and then solve it using the inverse matrix method. A second, and as I would point out, more efficient and general method is the Gauss-Jordan reduced row-echelon form (RREF) of an augmented matrix. 

We begin with three planes determined by the equations {x + y – 2z = 9; 2x – 3y + z = -2; and x + 3y + z = 2} This same system of equations can be expressed as the matrix equation.

Notice that the left matrix is made up of the coefficients of the three variable terms in each equation, and the right matrix contains the constant terms. We can find the intersection by taking the inverse of the left matrix and multiplying on the left of both sides of the equation. The simplified result gives

This seems to be the most commonly taught method, and the one that students and teachers seem to prefer, and yet it has two major disadvantages. The first disadvantage is that it tells you little or nothing about systems which have a solution, but not a single unique solution. In fact, it seems most students (or teachers?) can not distinguish between the cases (and there are several different ones) with no solutions from the ones with an infinite number of solutons. This same defect applies to attempts to use Cramers Rule. The second problem is that the inverse method is more computationally complex, that is, it takes more operations for the solution than the alternative RREF method, and the difference grows as problems reach higher orders of magnitude. For the problems that are generally assigned at the high school level, the difference in computability presents no real problem, but the difference in the range of applicable questions can be very significant in a students understanding of general systems of three equations.

In contrast with the Inverse method that will only work if the three planes intersect in a single point, the RREF form will allow us to work with systems which do not even have the same number of equations as unknowns. This is the type of situation created when we try to find the line of intersection of two planes.

RREF for two planes

So let's talk about the situation where two plains intersect in a point.  

We will use the equations 2x + 3y – 3z = 14 and –3x + y + 10z = -32. When we write an augmented matrix for the system of only two equations we get a 2x4 matrix, shown here:

When we reduce this system, by matrices or otherwise, we get

which is a matrix expression of x - 3z = 10 and y + z = -2.

Well, what can we do with that to help us find the line of intersection..... baby steps.... let's find one point on the intersection. 

 

We notice that both equations contain a z variable, it might occur to us to ask, “What happens if we substitute different values in for z?”. For example, if we try z=0 we note that from the first equation we get x=10 and from the second we get y=-2. What does this tell us about the point (10, -2, 0). If we check it against the two original equations we notice that the point makes both equations true, 2(10) + 3(-2) – 3(0) = 14 and –3(10) + (-2) + 10(0) = -32. So the point (10, -2, 0) is on both planes and therefore must lie on the line that is their intersection.

Can we find more points? What happens if we try z=1 or z=2 or other values. Using z=1 we get x – 3(1) = 10 which simplifies to x=13 ; and y +(1) = -2 which simplifies to y=-3. Checking the point (13, -3, 1) we see that it also makes both equations true, and so it must also be on the line of intersection.

And now we have come full circle, and we are back to the starting point, and now they really write the equation of a line through two points, in pretty much and dimension.  but what surprises may jump out in 4-D????? (cue dramatic music)

----(More to come)

BEWARE

Four Space Ahead

Addendum....  I first wrote this post around 2003 to 2005 (yeah, I write slowly.) and retired from formal education in 2011.  By now I am sure that all the suffering through New Math, the Agenda for Action, and  and Common Core, that the sad state of graduating HS seniors is no longer in the mess my generation left it, so please forgive me if this all seems unbelievably ancient to you.  Think of it as a reminder of how far Math has come on the shoulders of a New generation of scholar educators.   and b) really important. And I share with my students that it seems incredible to me that they, the best and brightest mathematics students in our school, are nearing the end of their high school education (many are seniors) and they can't do one of the most simple acts of coordinate geometry; that is, "Given two points, write the equation of a line containing the two points" (How does the standard for your school read?). 

 They look at me in wonder, confusion, and perhaps some doubt of my sanity. After all, we have done that thousands of times. I continue to bemoan their lack of ability until finally someone will challenge me...."But, Mr. B, we CAN do that. We do it all the time." Ok, We'll see, and I turn and write two points on the board... such as (3,1,2) and (2,4,3). They are so sure of their ability that they already have their pencils to paper when they realize they have no idea how to begin. I let them talk, explore, suggest ideas, and I wait, and I wait.... I have never had a student come up with an equation. Some will suggest it must be something like z=ax+by+c or something..... but NOT ONE ever hit upon a correct equation of the line in question.... Then we talk... Not about how to, that will come later, they will discover it on their own as a natural generalization, but about why not.. I admit to them that most of the students who graduate from high school (and in fact, many of the math teachers they have studied under) can not do this simple act of writing the equation of a line in the three space dimension that they live in. And I tell them that in the following weeks they will learn to do some of the simple geometry they know in the dimension they live in.

 I walk them through a simple vector approach to lines in the coordinate plane. We take y= 3x-1 and rewrite it as (x,y)= (0,-1) + t(1,3). Within minutes every kid in the class can write the equation of a line given two points in this vector form although a few struggle with the seeming reversal of order of the "slope"(in truth, several still mess up regularly when they try to use slope intercept, yet they seem reluctant to adopt the seemingly easier point-slope form). They are quickly taking two points and talking about "point vectors" and "slope vectors" as if they had used them forever. And each year it startles me anew that after a half-hour of an alternate approach, every student will intuitively generalize the method to produce a three-space equation of a line without any help...and then with a little faltering over the "fourth" variable, they can do the same thing in the barely imaginable four-space. Later we will write the equations of planes in space given three points and do some simple analytic geometry in three space. Many of them struggle with the idea of projections of lines and minor details, but I at least feel like I have made a small step to preparing them to function mathematically in the three-space they live in. And if the string-theory guys are right, and we really have a ten-dimensional universe.... no big deal, they can extend vectors to any dimension. But I wonder each year... why are we not introducing this more at an early age (alg I?). I will talk later about some of the advantages I see, and maybe you can tell me what I missing that would make it a bad idea. (Actually in the last few years I've been working a unit in parametric equations and vectors into my Alg II classes with lots of Bowditch curves (Lissajous?) and three D lines and planes and three D geometry ideas.)

I got this comment from one friend... 

Dave L. Renfro said...

Here's something marginally related to your topic. In 4 dimensional space it is possible for two planes to have exactly one point in common. A very simple example is the xy-plane, namely all points of the form (x,y,0,0) where x and y vary over the real numbers, and the zw-plane, namely all points of the form (0,0,z,w) where z and w vary over the real numbers. It is easy to see that these two planes have only the origin (0,0,0,0) in common, since x = y = z = w = 0 is the only solution to (x,y,0,0) = (0,0,z,w).

For what it's worth, I first learned about this possibility in a "side-bar diagram" in the following calculus book, which I was trying to work through during the 1974-75 school year (I managed to cover about the first 1/4 of the book), when I was in the 10th grade. Incidentally, the first college math class I took (an ODE course, about a year later) was from one of the authors (Embry) of this text.

Embry/Schell/Thomas, "Calculus and Linear Algebra. An Integrated Approach" (1972)

At this point you might try exploring other possible intersections between two four spaces, intersecting at a single point other than (0,0,0,0) , or a in a line, or a 3-plane.  Send me your best ideas, especially for sharing this with students.

This was all back in about 2009 and I retired two tears later and never followed up on this idea with a class, but I have some thoughts and will share them by adding to this post as time goes on, so check back...and share your ideas> I will pass them along as I go.  

I did come up with one more idea I wish I had thought of while I was teaching.  Somewhere in their HS math education most students who stay at it will learn to solve systems of equations and understand how to use matrices to find the point where three planes intersect IF they intersect in a single point.  (Thanks Dave!)

Planes in Space- part One

Take your basic bright kid in alg II or pre-calc, or often in calculus, and ask, "What is the intersection of two lines?"... They say, "A point."... good answer.

"Can you write the equation of a line?" Again they are on target.
"If I give you the equation of two lines on the plane; can you find their point of intersection?" The good ones can, and know they can.

Now ask the same bright kids, "What is the intersection of two planes in space?". They answer correctly again, "A line."

So far everything is great, but now we ask them to write the equation of a plane.... uhhh... gee..... and at this point, when asked about one of the fundamental structures of plane geometry, their analytic geometry skills are exhausted.

My experience was that after solving systems of three equations in three unknowns, they remembered how to do it (mostly) but didn't know what it was they had done???? 

but thankfully after a brief review a light goes on, and they understand.  

 Still, a very few may actually be able to produce x+y+z=1 or some other for the equation of a plane. Now we ask about the intersection of two planes, and almost none of them can do it. The scary part, is that very (very) few of the teachers of alg II and above that I have questioned about this could provide an answer either .

I would begin by recalling that an equation in three variables, such as 2x+3y+z=6 can represent a plane in space. When students had three such equations that intersected in a unique point, they found the solution by one of several methods. Most students learn to solve such equations by the methods called elimination and substitution at the very least. Others may have also been introduced to Cramer’s rule for solving systems with determinants and perhaps two methods using matrices.

The most commonly taught matrix method is to write a matrix equation and then solve it using the inverse matrix method. A second, and as I would point out, more efficient and general method is the Gauss-Jordan reduced row-echelon form (RREF) of an augmented matrix. 

We begin with three planes determined by the equations {x + y – 2z = 9; 2x – 3y + z = -2; and x + 3y + z = 2} This same system of equations can be expressed as the matrix equation.

Notice that the left matrix is made up of the coefficients of the three variable terms in each equation, and the right matrix contains the constant terms. We can find the intersection by taking the inverse of the left matrix and multiplying on the left of both sides of the equation. The simplified result gives

This seems to be the most commonly taught method, and the one that students and teachers seem to prefer, and yet it has two major disadvantages. The first disadvantage is that it tells you little or nothing about systems which have a solution, but not a single unique solution. In fact, it seems most students (or teachers?) can not distinguish between the cases (and there are several different ones) with no solutions from the ones with an infinite number of solutons. This same defect applies to attempts to use Cramers Rule. The second problem is that the inverse method is more computationally complex, that is, it takes more operations for the solution than the alternative RREF method, and the difference grows as problems reach higher orders of magnitude. For the problems that are generally assigned at the high school level, the difference in computability presents no real problem, but the difference in the range of applicable questions can be very significant in a students understanding of general systems of three equations.

In contrast with the Inverse method that will only work if the three planes intersect in a single point, the RREF form will allow us to work with systems which do not even have the same number of equations as unknowns. This is the type of situation created when we try to find the line of intersection of two planes.

RREF for two planes

So let's talk about the situation where two plains intersect in a point.  

We will use the equations 2x + 3y – 3z = 14 and –3x + y + 10z = -32. When we write an augmented matrix for the system of only two equations we get a 2x4 matrix, shown here:


When we reduce this system, by matrices or otherwise, we get

which is a matrix expression of x - 3z = 10 and y + z = -2.

Well, what can we do with that to help us find the line of intersection..... baby steps.... let's find one point on the intersection. 

 

We notice that both equations contain a z variable, it might occur to us to ask, “What happens if we substitute different values in for z?”. For example, if we try z=0 we note that from the first equation we get x=10 and from the second we get y=-2. What does this tell us about the point (10, -2, 0). If we check it against the two original equations we notice that the point makes both equations true, 2(10) + 3(-2) – 3(0) = 14 and –3(10) + (-2) + 10(0) = -32. So the point (10, -2, 0) is on both planes and therefore must lie on the line that is their intersection.

Can we find more points? What happens if we try z=1 or z=2 or other values. Using z=1 we get x – 3(1) = 10 which simplifies to x=13 ; and y +(1) = -2 which simplifies to y=-3. Checking the point (13, -3, 1) we see that it also makes both equations true, and so it must also be on the line of intersection.

And now we have come full circle, and we are back to the starting point, and now they really write the equation of a line through two points, in pretty much and dimension.  but what surprises may jump out in 4-D????? (cue dramatic music)

----(More to come)

BEWARE

Four Space Ahead

Addendum....  I first wrote this post around 2003 to 2005 (yeah, I write slowly.) and retired from formal education in 2011.  By now I am sure that all the suffering through New Math, the Agenda for Action, and  and Common Core, that the sad state of graduating HS seniors is no longer in the mess my generation left it, so please forgive me if this all seems unbelievably ancient to you.  Think of it as a reminder of how far Math has come on the shoulders of a New generation of scholar educators.