Question

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

Answer 1

The correct answer is:$=\frac{1}{3}$
Here, X represents the number of sixes obtained when two dice are thrown simultaneously. Therefore, X can take the value of 0, 1, or 2.
∴ P (X = 0) = P (not getting six on any of the dice)$=\frac{25}{36}$
P (X = 1) = P (six on first die and no six on second die) + P (no six on first die and six on second die)
$=2(\frac{1}{6}\times \frac{5}{6})=\frac{10}{36}$
P (X = 2) = P (six on both the dice)$=\frac{1}{36}$

Therefore, the required probability distribution is as follows.X	0	1	2
P(X)	$\frac{25}{36}$	$\frac{10}{36}$	$\frac{1}{36}$

Then, expectation of X$=E(X)=\sum X_iP(X_i)$
$=0 \times \frac{25}{36}+1\times \frac{10}{36}+2\times \frac{1}{36}$
$=\frac{1}{3}$