How many sixes on an 18-face die averaging 4

Question

A fair die has $18$ faces. Each face is marked with either a $3$ or a $6$ , and the mean value of a throw equals $4$ . Determine the number of faces marked $6$ .

Short answer

6. With $n$ faces showing $6$ , the mean is $\tfrac{6n+3(18-n)}{18}=3+\tfrac{n}{6}$ . Setting this equal to $4$ gives $n=6$ .

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Lessons behind this question

Counting and Combinatorics
Expectation, Mean, and Variance
Conditional Expectation and Variance Revisited

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