A die is thrown $120$ times and getting $1$ or $5$ is considered a success. Find the mean and variance of the number of successes.
Probability of getting a $1$ when a dice is thrown $= \dfrac{1}{6}$
Probability of getting a $5$ when a dice is thrown $= \dfrac{1}{6}$
Let $p = \;$ probability of getting a $1$ or a $5$ when a dice is thrown $= \dfrac{1}{6} + \dfrac{1}{6} = \dfrac{1}{3}$
$\therefore \;$ $q = 1 - p = 1 - \dfrac{1}{3} = \dfrac{2}{3}$
Number of times the dice is rolled $= n = 120$
By definition, Mean $= n \times p$
$\therefore \;$ $\text{Mean } = 120 \times \dfrac{1}{3} = 40$
By definition, Variance $= n \times p \times q$
$\therefore \;$ $\text{Variance } = 120 \times \dfrac{1}{3} \times \dfrac{2}{3} = \dfrac{80}{3}$