Two dice are thrown simultaneously. If $X$ denotes the number of sixes, find the expectation of $X$

.: Sample space $S = \{ 1,2,3,4,5,6\}$

Let $p$ be the probability of getting
${\mathop{\rm six}\nolimits} = \cfrac{1}{6}$

and $q$ be the probability of not getting ${\mathop{\rm six}\nolimits} = 1 - \cfrac{1}{6} = \cfrac{5}{6}$

since $X$ denotes the random variable which represents the number of sixes,

$\therefore$ $X$ can assume the values 0, 1, 2
$\therefore$

$P(X = 0) = q \times q = \cfrac{5}{6} \times \cfrac{5}{6} = \cfrac{{25}}{{36}}$

$P(X = 1) = p \times q + q \times p = \cfrac{1}{6} \times \cfrac{5}{6} + \cfrac{5}{6} \times \cfrac{1}{6} = \cfrac{{10}}{{36}} = \cfrac{5}{{18}}$

and $P(X = 2) = p \times p = \cfrac{1}{6} \times \cfrac{1}{6} = \cfrac{1}{{36}}$

Hence, the probability distribution is :

Hence, the expectation of

$X = E(X) = \sum x p(x) = 0 \times \cfrac{{25}}{{36}} + 1 \times \cfrac{5}{{18}} + 2 \times \cfrac{1}{{36}}$

$= 0 + \cfrac{5}{{18}} + \cfrac{2}{{36}} = \cfrac{{12}}{{36}}$