Suppose a dice is thrown $8$ times and success is considered as obtaining either a $5$ or $6$. What is the variance of the number of successes?
Attempt: Let the indicator variable $X_i$ be $1$ when either a $5$ or a $6$ turns up on the dice on the $i$th throw and let $X_i$ be $0$ otherwise.
Then: Expected number of successes = $\sum E(X_i)=8 \cdot P(\text{Success}) = 8 \cdot \dfrac{1}{3}$
If $X$ is the total number of success, $E(X^2)= \sum 1^2 \cdot \dfrac{1}{3}=\dfrac {8}{3}$
$Variance = \dfrac{8}{3}- \dfrac{8^2}{3^2} <0$
Where is the error?
Thanks for the help!