Question

Two dice are thrown, simultaneously. If $X$ denotes the number of sixes, then the expected value of $X$ is

• A. $E(X)=\frac{1}{3}$
• B. $E(X)=\frac{2}{3}$
• C. $E(X)=\frac{1}{6}$
• D. $E(X)=\frac{5}{6}$

Answer 1

Answer: (A)

$X$ can take values $0,1,2$.
$P(X=0)=$ Probahility of not getting six on any dice $=\frac{25}{36}$
$P(X=1)=$ Probability of getting one six $=\frac{10}{36}$
$P(X=2)=$ Probability of getting two six $=\frac{1}{36}$
Thus, the probability distribution is

X	0	1	2
P(X)	$\frac{25}{36}$	$\frac{10}{36}$	$\frac{1}{36}$

$\therefore E(X)=0\times \frac{25}{36}+1\times \frac{10}{36}+2\times \frac{1}{36}$
$=\frac{10}{36}+\frac{2}{36}=\frac{1}{3}$