Second die roll beating the first

Question

A fair six-sided die is thrown two times in a row. What is the chance that the second throw shows a strictly larger number than the first?

Short answer

$5/12$ . The two throws match with probability $1/6$ . Otherwise one of them is larger, and by symmetry each throw is equally likely to be the larger one, so the answer is $\tfrac12\left(1-\tfrac16\right)=\tfrac{5}{12}$ .

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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