Question

Two die are thrown simultaneously. If X denote the number sixes, find the expectation of X.

Answer 1

Let X be the random variable which denotes the number of sixes on two dice. So, X may have values 0,1 or 2.

When a dice is rolled once, probability of obtaining six $=\frac{1}{6}$ and probability of obtaining a non-six $=\frac{5}{5}$

P(X =0) =P (non-six on both dice)=$\frac{5}{6}\times \frac{5}{6}=\frac{25}{36}$

P(X=1)=P (six on first and non -six on the second)+P(non-six on the first and six on the second)

$=\frac{1}{6}\times \frac{5}{6}+\frac{1}{6}\times \frac{5}{6}=\frac{10}{36}$

P(X=2)=P(six on both the dice) $=\frac{1}{6}\times \frac{1}{6}=\frac{1}{36}$

Therefore, the required probability distribution is as follows:

$\begin{array}{cccc}X& 0& 1& 2\\ P\left(X\right)& \frac{25}{36}& \frac{10}{36}& \frac{1}{36}\end{array}$

$\therefore$ Expectation of X=mean of the variable X

=$\sum XP\left(X\right)=0\times \frac{25}{36}+1\times \frac{10}{36}\times +2\times \frac{1}{36}=\frac{12}{36}=\frac{1}{3}.$