Chance the smallest of four dice is at least three

Question

You roll four fair six-sided dice and look at the smallest number showing. What is the probability that the smallest number is at least 3, that is, every die shows 3 or more?

Accepted Answer

The smallest die is at least 3 exactly when all four dice individually land on 3, 4, 5, or 6. Each die does that with probability \frac{4}{6} = \frac{2}{3}, and the dice are independent, so P(	ext{min} \ge 3) = \left(\frac{2}{3}ight)^4 = \frac{16}{81} \approx 0.198. The general rule: the minimum of several dice is at least v precisely when every die is at least v, which multiplies to \left(\frac{m-v+1}{m}ight)^k for k dice with m faces. Takeaway. A question about a minimum being large is really an "all of them clear the bar" question, so the probabilities simply multiply.

Chance the smallest of four dice is at least three

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