Thursday 13 November 2008

Some additional History on the Problem of Points

After my recent post on the probability history I got a nice note from Jim Kiernan advising me about an article he posted in the March, 2001 issue of The Mathematics Teacher. He is (or at least was according to the article) a teacher at the Edward R. Murrow High School in Brooklyn, NY. His interests were listed as math history, and in particular, the origins of probability and statistics.

The entire article is well worth the read, and if you are also interested in math/stats history, it is worth the trip just for the references. In particular, and I suspect that a stats-history buff like Jim already knows, but one of the references, A. W. F. (Tony) Edwards, not only wrote a really neat book on "Pascal's Arithmetical Triangle, The Story of an Idea", but he was also the last student under the tutelage of the great statistician, R. A. Fisher at Cambridge. My personal appreciation to Professor Edwards include thanks for a guided tour around the great hall at Gonville and Caius ( pronounced "keys") to view the stained glass tribute to John Venn (and several other math/science people), and for giving me directions to the (totally hidden in vines at the time) grave of John Venn (photo at top). In his book on Pascal, Professor Edwards points out that it was the problem of points that prompted Pascal to write his famous treatise.

I have taken the liberty of copying a few key remarks from Mr. Keirnan's article that compliment the previous post, and am grateful to Jim for the note.

As early as the twelfth century, the Arabs were acquainted with the binomial triangle and used it to solve problems that involved combinations. Islamic tradition also deals with problems of dividing inheritances. Tartaglia repeated the tradition that Leonardo of Pisa (ca. 1200), commonly known as Fibonacci, was responsible for bringing the practice of algebra to Italy from Arabia. "Although no mention of the "problem" is attributed to Leonardo, its origins apparently also lie with the Arabs. Oystein Ore refers to an Italian manuscript, dating from approximately 1380, that is probably of Arab origin and that contains the "problem." The Arabs seem to have had all the right tools, but no record of a solution exists.

Tartaglia and Cardano both tried (and failed) to solve the problem and he includes their wrong answers in the article. Then, Several other futile attempts were made to solve the problem before it fell into obscurity. Galileo wrote about probability, but no extant version of the problem appears in his papers. Widespread knowledge of the binomial triangle existed throughout Europe. It appears in the works of Cardano, Tartaglia, and Mersenne. Yet no record exists of anyone's applying it to the problem. Finally, during the summer of 1654, the problem was solved in three different ways as the result of a correspondence between two of the greatest French mathematicians of the seventeenth century: Blaise Pascal and Pierre de Fermat.

The correspondence began in response to a pair of questions submitted by the Chevalier de Mere. The second of these problems would be the catalyst for the founding of probability theory. The first letter from Fermat "on division" is missing, but Pascal (1952, p. 475) responded on 29 July that the "method is very reliable and is the first that had occurred to me." Pascal claims to have found a "different method much shorter and simpler." The letter ends with the heartwarming observation that "truth is the same at Toulouse and at Paris."

Pascal's first method can best be explained using the ideas of recursion and weighted averages. When a total of three games is required to win, he considers three cases: (2, 1), (2, 0), and (1, 0). The first case, (2, 1), is a simple example; the second, (2, 0), gives the answer to Pacioli's problem; and the third, (1, 0), gives the answer to de Mere's problem. In each case, a total of 32 pistoles is wagered by each player. This number seems to have been selected so that the solution would be a simple ratio.

Analyzing the simple case, "they now play a game ... if the first player wins, he wins all the money ... if the second player wins each should withdraw his own stake" (Pascal 1952, p. 475). The result is a split of either [64; 01 or [32; 32]. The first player is "sure of having 32... as for the other 32 ... let us share equally." So if the game is interrupted before the next round, the correct split should be [48; 16]. The second case reverts to the first case when the second player wins the next game. If the game is interrupted at (2, 0), the player who has two games should get 48 pistoles plus half of 16. So the correct split for this case and Pacioli's problem would be [56; 81, or 7 : 1 in simplest form.

De Mere's problem requires finding "the value ... when two players are playing for three games and ... one player has only one game and the other none" (Pascal 1952, p. 475). Using the process of recursion developed so far brings the situation back to the previous case. If the first player wins, the status becomes (2, 0), which entitles him to 56 pistoles. If the first player loses, the status is even, (1, 1), which entitles him to 32. So in the case of an interruption, the first player should get 32 plus half of (56 - 32). The correct split is [44; 20].

Pierre de Fermat's solution, dated 24 August, depended on determining the number of games required to declare a winner. If player 1 needs m games more and player 2 needs n games more to win, then a winner must be declared after m + n - 1 more games. Fermat then listed all possible outcomes for four more games

and formed the ratio of wins by each player where a is a win for player 1 and b is a win for player 2: aaaa 1 abaa 1 baaa 1 bbaa 1 aaab 1 abab 1 baab 1 bbab 2 aaba 1 abba 1 baba 1 bbba 2 aabb 1 abbb 2 babb 2 bbbb 2

"Therefore, they must share the sum in the ratio of 11 to 5" (Pascal 1952, p. 475).

This result is equivalent to Pascal's solution [44; 20]. Gilles de Roberval, a member of Pascal's intellectual circle in Paris, was not pleased with this means of listing outcomes. He criticized the use of four games when two or three would determine a win.

The last of the three methods used to solve this problem is contained in Pascal's Treatise on the Arithmetic Triangle, which was written in 1654 but not published until 1665.

Thanks again Jim...

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