Origins of The Modern Theory of Probability
Around the year 1605, the founder of modern science, Galileo Galilee, was asked by his
patron, the Grand Duke Cosmo II, for help in solving the following gambling problem:
“Three dice are thrown: … long observation has made dice players consider (sum) ten to be
more advantageous than (sum) nine. Why?”
Proceeding as Cardano had indicated, Galileo listed all the 6
3
(=216) possible combinations of 3
dice, and then listed those combinations that produce 9 when summed, and those combinations
that produce 10 when summed. He showed that there were 25 combinations which summed to 9,
and 27 combinations that summed to 10.
5
Thus
Pr (sum 9) = 25/216 Pr (sum 10) = 27/216.
In a circuit of 216 throws, 3 dice will sum 10 more often than they sum 9 (on the average),
confirming the Grand Duke’s suspicion.
Another 17
th
century gambling enthusiast, the Chevalier de Mere (1607-1684), would bet
even money that he could get at least one six in every four rolls of a die. This seemed counter-
intuitive because we would expect, on the average, one six in every six throws of a die. Thus, de
Mere was able to entice many to bet against him. But de Mere’s conjecture proved correct, and
he won a considerable amount of money on the wager.
What is the probability that 6 will turn up in four throws of a six sided die? Let 6
1
= 6 on
throw 1 and 6
n
= 6 on throw n. Then it is tempting to represent this problem as follows:
Pr(6
1
or 6
2
or 6
3
or 6
4
) = Pr(6
1
) + Pr(6
2
) + Pr(6
3
) + Pr(6
4
) =
1/6 + 1/6 + 1/6 + 1/6 = 4/6 = 2/3.
5
For 9, (1,2,6) appears 6 times,(1,3,5)appears 6 times, (1,4,4) appears 3 times, (2,2,5) appears 3 times, (2,3,4)
appears 6 times, (3,3,3) appears 1 time. So throwing a total of 9 can appear 25 times in all; For 10, (1,3,6) appears 6
times,(1,4,5)appears 6 times, (2,4,4) appears 3 times, (2,2,6) appears 3 times, (2,3,5) appears 6 times, (3,3,4) appears
3 time. So throwing a total of 10 can appear 27 times in all. Therefore, the chance of throwing a total of 9 with three
fair dice was less than that of throwing a total of 10.