At least one low face among four dice

Question

You roll four fair six-sided dice. What is the probability that the smallest number showing is at most 2, that is, at least one die shows a 1 or a 2?

Accepted Answer

Rather than count the many ways a low face can appear, flip to the complement. "The minimum is at most 2" fails only when every die is 3 or more: P(	ext{min} \ge 3) = \left(\frac{4}{6}ight)^4 = \frac{16}{81}. Therefore P(	ext{min} \le 2) = 1 - \frac{16}{81} = \frac{65}{81} \approx 0.802. Takeaway. For "the minimum is small" events, take one minus the clean "all dice are large" probability.

At least one low face among four dice

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