# A pair of honest dice is rolled once. Find the expected value of the sum of the

A pair of honest dice is rolled once. Find the expected value of the sum of the two numbers rolled.
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Below is sample space for sum of pair of dice:
$S=\left\{2,3,4,5,6,7,8,9,10,11,12\right\}$
Evaluating probability of each event:
$\begin{array}{|cc|}\hline Sum& Probability\\ 2& 1/36\\ 3& 2/36\\ 4& 3/36\\ 5& 4/36\\ 6& 5/36\\ 7& 6/36\\ \hline\end{array}$
We know Expected value (E) is given by:
$E=\left({p}_{1}\cdot {x}_{1}\right)+\left({p}_{2}\cdot {x}_{2}\right)+\left({p}_{3}\cdot {x}_{3}\right)+\dots +\left({p}_{N}\cdot {x}_{N}\right)$
Evaluating expected value (E) of the sum of the two numbers rolled:
$E=\left(2\cdot \frac{1}{36}\right)+\left(3\cdot \frac{2}{36}\right)+\left(4\cdot \frac{3}{36}\right)+\left(5\cdot \frac{4}{36}\right)+\left(6\cdot \frac{5}{36}\right)+\left(7\cdot \frac{6}{36}\right)$
$=\frac{2+6+12+20+30+42+40+36+30+22+12}{36}$
$=\frac{252}{36}$
$=7$