Question

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

Answer 1

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.
$\therefore P(X=0)=P\left(notgettingsixonanyofthedice\right)=\frac{25}{36}$
$P(X=1)=P\left(sixonfirstdieandnosixonseconddie\right)+P\left(nosixonfirstdieandsixonseconddie\right)$
$=2\left(\frac{1}{6}\times \frac{5}{6}\right)=\frac{10}{36}$
$P(X=2)=P\left(sixonboththedice\right)=\frac{1}{36}$
Therefore, the required probability distribution is as follows.
Then, expectation of $X=E\left(X\right)=\sum X_{i}P\left(X_i\right)$
$=0\times \frac{25}{36}+1\times \frac{10}{36}+2\times \frac{1}{36}$
$=\frac{1}{3}=0.33$