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Largest of $2$ rolls

ProbabilityEasy

A fair $6$ -sided die is rolled $2$ times independently. What is the probability that the larger of the $2$ results is at most $4$ ? Give a fraction in lowest terms.

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Hint

The maximum is small only if every single roll is small.

Solution

The larger result is at most $4$ exactly when both rolls are at most $4$ .
Each roll is at most $4$ with probability $\frac{4}{6} = \frac{2}{3}$ .
By independence the probability is $(\frac{2}{3})^{2} = \frac{4}{9}$ .
The answer is $\frac{4}{9}$ .

Answer: $\frac{4}{9}$

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