Three successive dice coming up strictly ascending

Question

Three fair dice are rolled one after another. Find the probability that the first is less than the second and the second is less than the third.

Short answer

$5/54$ . The three values must be distinct, and each set of three distinct faces appears in increasing order in exactly one of its arrangements. So the count is $\binom63=20$ out of $216$ outcomes.

Adapted from Zhou, A Practical Guide to Quantitative Finance Interviews, sec. 4.3 Conditional Probability and Bayes' Formula, 'Dice order', p.77; 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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