Question

A dice is rolled 5 times. Find the probability of getting 4 exactly 3 times.

Answer 1

The probability of getting a '4' exactly 3 times when a dice is rolled 5 times is $\frac{125}{3888}$.

Answer 2

This is a binomial probability problem. The formula is $P(X=k) = C(n, k) \cdot p^k \cdot (1-p)^{n-k}$. Here, $n=5$ (number of rolls), $k=3$ (number of times '4' appears), $p=1/6$ (probability of rolling a '4'). $C(5, 3) = \frac{5!}{3!2!} = 10$. $P(X=3) = 10 \cdot (\frac{1}{6})^3 \cdot (\frac{5}{6})^{5-3} = 10 \cdot \frac{1}{216} \cdot \frac{25}{36} = \frac{250}{7776} = \frac{125}{3888}$.