Monday 10 October 2022

Some Notes on the Birthday Problems


A couple of email questions asked about the birthday problems... One questioned the assumption that births are not uniformly distributed in the months (or days of the month)... which is quite true, and worth backing up with some info, but it actually makes the probability of a match MORE likely at n=23 than it would be if the births were uniformly distributed.
Math Trek article by Ivars Peterson has a table of monthly probabilities showing the daily frequency of birth each month. September seems to be the most popular month, but the differences are almost negligible in the total probability calculation.

A greater difference is due to the fact that in modern times, far fewer people are born on a weekend. Induced labor saves many doctors from a spoiled yachting weekend. The Fathom Graph below shows the distribution of birthdays for births in the U.S. in 1978. It was used by Professor Geoffrey Berresford in his article: "The uniformity assumption in the birthday problem, Math. Mag. 53 1980, no. 5, 286-288." If you plot a times series of the data you will have a nice example of periodic data. The Saturdays and Sundays show up well below the others, (yearday.jpg)... The actual data can be found at the Chance Data Base at Dartmoth.

One more graph, this one from the Skeptical Inquirer on line magazine. It relates to the probability of a match or near match (same day or one day apart) with n people. The curve shows the probability of a match on the vertical axis, and the number of people on the horizontal. The dots are for the traditional problem, and the solid line is the "near match" probability.

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