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Maximum stays low

ProbabilityEasy

Roll $3$ fair $6$ -sided dice independently. What is the probability that the largest value shown is at most $4$ ?

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Hint

The maximum is at most the bound exactly when every die is at most the bound.

Solution

The largest value is at most $4$ exactly when each of the $3$ dice shows at most $4$ .
For one die this has probability $\frac{4}{6} = \frac{2}{3}$ .
The dice are independent, so the probability is $(\frac{2}{3})^{3}$ .
This equals $\frac{8}{27}$ .

Answer: $\frac{8}{27}$