A history of the mathematical theory of probability from the time of Pascal to that of Laplace / by I. Todhunter.

  • Isaac Todhunter
Date:
1865
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Licence: Public Domain Mark

Credit: A history of the mathematical theory of probability from the time of Pascal to that of Laplace / by I. Todhunter. Source: Wellcome Collection.

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    seventeen can each be thrown in three ways ; and so on. Cardan also gives the following list of the number of cases in Fntillo : 1 2 3 4 5 6789 10 11 12 108 111 115 120 126 133 33 36 37 36 33 26 Here we have corrected two misprints by the aid of Cardan’s verbal statements. It is not obvious what the table means. It might be supposed, in analogy with what has already been said, that if a person throws three dice he is to be considered to have thrown a given number when one of the dice alone exhibits that number, or when two dice together exhibit it as their sum, as well as when all the three dice exhibit it as their sum: and this would agree with Cardan’s remark, that for numbers higher than twelve the favourable cases are the same as those already given by him for three dice. But this meaning does not agree with Cardan’s table; for with this meaning we should proceed thus to find the cases favourable for an ace: there are 53 cases in which no ace appears, and there are 63 cases in all, hence there are 63 — 53 cases in which we have an ace or aces, that is 91 cases, and not 108 as Cardan gives. The connexion between the numbers in the ordinary mode of using dice and the numbers which Cardan gives appears to be the following. Let n be the number of cases which are favour- able to a given throw in the ordinary mode of using three dice, and N the number of cases favourable to the same throw in Cardan’s mode; let m be the number of cases favourable to the given throw in the ordinary mode of using two dice. Then for any throw not less than thirteen, 17= n ; for any throw between seven and twelve, both inclusive, 17= 3m + n; for any throw not greater than six, I\ = 108 -f- 3in -f- n. There is only one deviation from this law; Cardan gives 26 favourable cases for the throw twelve, and our proposed law would give 3 + 25, that is 28. We do not, however, see what simple mode of playing with three dice can be suggested which shall give favourable cases agreeing in number with those determined by the above law. 6. Some further account of Cardan’s treatise will be found 1—2
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