Three dice with a maximum of exactly four

Question

Three fair six-sided dice are thrown. What is the probability that the largest number showing is $4$ ?

Short answer

$37/216$ . All three dice are at most $4$ with probability $(4/6)^3$ and at most $3$ with probability $(3/6)^3$ . The difference is $\tfrac{64-27}{216}=\tfrac{37}{216}$ .

Adapted from QuantQuestions.io; question and answer rewritten by SelfMadeQuant.com.

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Lessons behind this question

Counting and Combinatorics
Expectation, Mean, and Variance
Sample Spaces, Events, and the Probability Axioms

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