De Méré and Galileo
Two gambling puzzles from the 1600s where the gamblers' instincts, sharpened by thousands of games, noticed differences of about one or two percentage points.
From the slides: Why should anyone care about chance? and Probability and what happens when you learn something.
The Chevalier's two bets (1654)
Bet A: roll one die four times; win if at least one six appears. Bet B: roll two dice 24 times; win if at least one double six appears. The Chevalier reasoned that 4 × 1/6 = 24 × 1/36 = 2/3, so the bets should be the same. He kept winning on A and losing on B.
Galileo's three dice
With three dice, a total of 9 and a total of 10 can each be made in six ways (1+2+6, 1+3+5, …). Yet Italian gamblers noticed that 10 turns up a little more often. Counting the 216 ordered outcomes gives P(9) = 25/216 and P(10) = 27/216.
What to notice
- Bet A is a small winner and Bet B a small loser. Early on either can look better; it takes a few thousand games for the gap of 2.6 percentage points to show clearly. De Méré had played enough to feel it.
- “Four chances of 1/6” is not 4/6: the chances overlap. The complement gives the right answer: P(no six in four rolls) = (5/6)4.
- Galileo's point: the six “ways” to make 9 are not equally likely. 1+2+6 can come up in 3! = 6 orders, 3+3+3 in only one. Always count outcomes that really are equally likely.