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

Toolbox:
• Mean of the probability distribution = $\sum (X_i \times P(X_i))$
If two dice are thrown simultaneously, the sample space can be depicted as follows; S = $\begin{Bmatrix} 1,1 & 1,2&1,3&1,4&1,5&1,6 \\ 2,1 & .... &\\ 3,1 &...\\ 4,1 &....\\ 5,1 & ...\\ 6,1 & 6,2 & 6,3 & 6,4 & 6,5 & 6,6 \end{Bmatrix}$
Let X be the random variable for our problem. X can assume the values of 0, 1 and 2.
P (X = 0) = P (no sixes) = $\begin{Bmatrix} 1,1 & 1,2&1,3&1,4&1,5 \\ 2,1 & .... &\\ 3,1 &...\\ 4,1 &....\\ 5,1 & ...\\ \end{Bmatrix}$ = $\large\frac{25}{36}$
P (X = 1) = P ( 6 on 1st die and no 6 on 2nd OR 6 on 2nd die and no 6 on 1st) = $\begin{Bmatrix} 1,6 & 2,6&3,6&4,6&5,6 \\6,1 & 6,2 & 6,3 & 6,4 & 6,5 \end{Bmatrix} = \large\frac{10}{36}$
P (X = 2) = P (2 sixes) = {$6,6$} = $\large\frac{1}{36}$
Given the above distribution, the mean can be calculated as follows:
$\sum (X_i \times P(X_i)) = 0 \times \large\frac{25}{36}$$\;+ 1 \times \large\frac{10}{36}$ $\;+2\times\large\frac{1}{36} = \frac{12}{36} = \frac{1}{3}$