Three dice all showing at least four

Question

You roll three fair six-sided dice. What is the probability that the smallest number showing is at least 4, that is, every die shows 4, 5, or 6?

Accepted Answer

The minimum is at least 4 exactly when all three dice land on 4, 5, or 6. Each die does so with probability \frac{3}{6} = \frac{1}{2}, and the rolls are independent, so P(	ext{min} \ge 4) = \left(\frac{1}{2}ight)^3 = \frac{1}{8} = 0.125. Takeaway. "The minimum is at least v" means every die clears v, so just raise the single-die chance to the power equal to the number of dice.

Three dice all showing at least four

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