At Least One Six

Question

You roll a fair die four times. What is the probability you see at least one 6?

Accepted Answer

Work with the complement. The probability a single roll is not a 6 is frac56, and rolls are independent: P(textno six in 4 rolls) = left(frac56right)^4 = frac6251296. Therefore P(textat least one 6) = 1 - frac6251296 = frac6711296 approx 0.5177. Slightly better than a coin flip. This is de Méré's classic gambling problem from 1654, he made money betting on it at even odds. Speed check Mental approximation: (5/6)^4 = (1 - 1/6)^4 approx e^-4/6 approx e^-0.667 approx 0.513, so the answer is roughly 0.49… the exponential approximation slightly overshoots the decay; exact is 0.518. Knowing left(1-frac1nright)^k approx e^-k/n is a workhorse for fast estimates.

At Least One Six

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