Top three of five dice summing to 18

Question

Five fair dice are thrown, and you add up the three highest values shown. What is the probability that this total equals $18$ ?

Short answer

$\tfrac{23}{648}$ . The three highest sum to 18 only if they are all sixes, i.e. at least three of the five dice show 6. The count of outcomes is $\binom53 5^2+\binom54 5+1=276$ out of $6^5=7776$ , giving $\tfrac{23}{648}$ .

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

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Lessons behind this question

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

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