A fair die is rolled five times. What is the probability of obtaining exactly three sixes?

Step-by-step solution

1. Use the binomial probability formula: $P(k)=\left(\genfrac{}{}{0ex}{}{n}{k}\right)p^k(1-p{)}^{n-k}$.
2. Here $n=5$, $k=3$, and $p=\frac{1}{6}$ for a six.
3. Compute the binomial coefficient: $\left(\genfrac{}{}{0ex}{}{5}{3}\right)=10$.
4. Compute $p^k=(\frac{1}{6}{)}^3=\frac{1}{216}$.
5. Compute $(1-p{)}^{n-k}=(\frac{5}{6}{)}^2=\frac{25}{36}$.
6. Multiply: $10\times \frac{1}{216}\times \frac{25}{36}=\frac{250}{7776}$.
7. Reduce the fraction: divide numerator and denominator by 2 to get $\frac{125}{3888}$.

Concept tested: Binomial distribution probability

Published 18 September 2026 · How this solution was prepared and checked