Newton–Pepys problem
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The Newton–Pepys problem is a probability problem concerning the probability of throwing sixes from a certain number of dice.[1]
In 1693 Samuel Pepys and Isaac Newton corresponded over a problem posed by Pepys in relation to a wager he planned to make. The problem was:
- Which of the following three propositions has the greatest chance of success?
-
- A. Six fair dice are tossed independently and at least one “6” appears.
- B. Twelve fair dice are tossed independently and at least two “6”s appear.
- C. Eighteen fair dice are tossed independently and at least three “6”s appear.[2]
Pepys initially thought that outcome C had the highest probability, but Newton's analysis showed that outcome A actually has the highest probability.
[edit] Solution
The probabilities of outcomes A, B and C are:[1]
These results may be obtained by applying the binomial distribution (although Newton obtained them from first principles). In general, if P(N) is the probability of throwing at least n sixes with 6n dice, then:
As n grows, P(N) descreases monotonically towards an aymptotic limit of 1/2.
[edit] References
- ^ a b Eric W. Weisstein, Newton-Pepys Problem at MathWorld.
- ^ Isaac Newton as a Probabilist, Stephen Stigler, University of Chicago
