So let $A$ be the event the die falls either "one", "two", or "three"
Let $B$ be the event the die falls either "four" or "five"
and let $C$ be the event the die falls "six"
$P(A) = \frac{1}{2}$, $P(B) = \frac{1}{3}$, $P(C) = \frac{1}{6}$
So since $X$ is the amount Betty receives, to find $E(X)$...
$E(X) = \frac{1}{2} * 2 + \frac{1}{3} * 3 + \frac{1}{6} * 6 = 3$
and to find the variance:
$var(x) = (2-3)^{2} * \frac{1}{2} + (3-3)^{2} * \frac{1}{3} + (6-3)^{2} * \frac{1}{6} = 2$
Is my solution correct?