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Median of $3$ rolls

StatisticsHard

A fair $6$ -sided die is rolled $3$ times independently. What is the probability that the median of the $3$ results equals $3$ ? Give a fraction in lowest terms.

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Hint

Write $P (median = 3)$ as $P (median \leq 3)$ minus $P (median \leq 2)$ .

Solution

The median is at most $3$ exactly when at least $2$ of the $3$ rolls are at most $3$ ; by symmetry this has probability $\frac{1}{2}$ .
The median is at most $2$ when at least $2$ rolls are at most $2$ , each with probability $\frac{1}{3}$ : $3 \times (\frac{1}{3})^{2} \times (\frac{2}{3}) + (\frac{1}{3})^{3} = \frac{7}{27}$ .
Subtract: $\frac{1}{2} - \frac{7}{27} = \frac{13}{54}$ , so the answer is $\frac{13}{54}$ .

Answer: $\frac{13}{54}$

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