Review1 of Liber De Ludo Aleae (Book on Games of Chance) by ...

The above sequence <strong>of</strong> expected values conforms to Pascal’s rule.
To illustrate, Pascal considers a game in which a player has obtained 1 point and needs 4 more. He notes
that at most 8 plays would be required to complete the game (either player A throws 4 more points, or
player B will throw the required 5). He observes that 1/2 <strong>of</strong> the number <strong>of</strong> combinations <strong>of</strong> 4 from 8,
divided by a sum consisting <strong>of</strong> this same value, plus the combinations <strong>of</strong> 5,6,7 and 8 from 8, gives the
same proportion as 1/2 . 3/4 . 5/6 . 7/8 = 35/128.
This is the case, since in general:
1/2 . 3/4 . 5/6 . … . (2n-1)/(2n) = (2n-1)!/n!(n-1)! . 1/2 2n-1 , with:
(2n-1)!/n!(n-1)! = 1/2 . (2n!/n!n!), and
2 2n-1 = 1/2 . (1+1) 2n , and
1/
2 ⋅ ( 1+
1)
2n
= 1/
2 ⋅
2n
∑
i=
0
⎛2n
⎞
⎜ ⎟
⎝i
⎠
In the July 29 th letter, Pascal also provides two tables indicating a division <strong>of</strong> wagers for games <strong>of</strong> dice
suspended at different stages. The tables are not accompanied with detailed explanations. Pascal also
relates the observations, and questions <strong>of</strong> Monsieur de Mere, relating to a game <strong>of</strong> dice requiring (at least)
one six to turn up in 4 throws. The odds given in favour <strong>of</strong> this event are 671 to 625. Again, the
computations are not provided, however, they correspond to the probability given by:
∑
i=
i
⎟
4 4
⎜ ( 1/
6)
( 5 / 6)
1 ⎝ ⎠
⎛ ⎞ i 4−i
Also, it is noted that there is a “disadvantage” in throwing two sixes in 24 such plays. Using the above
formula, summing the combinations <strong>of</strong> 1,2,…24 out <strong>of</strong> 24, with associated probabilities 1/36 and 35/36
(to the appropriate exponents) the probability for the event can be shown to be about 0.49. That Monsieur
de Mere noticed in practice this “disadvantage” is remarkable (he must have observed, and / or played,
many such games).
The remaining correspondence includes an interesting dispute over the interpretation <strong>of</strong> combinations <strong>of</strong>
events, used as a means for computing equitable settlements for wagers (establishing the proportion <strong>of</strong>
funds to be distributed).
However, for our purposes, at this stage, it is sufficient to appreciate that combinatorial methods, and the
identification <strong>of</strong> equipossible events, are the cornerstones for the emerging theory <strong>of</strong> probability, with
early applications for binomial expansions. Instead <strong>of</strong> using the terms “chance” or “probability”, our
correspondents used words such as “favour” or “advantage” and “disadvantage”, which convey the same
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