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

Solution:

A die is tossed twice in this game.

When a dice is thrown twice, the total number of observations is (6 \times 6)=36. Let \mathrm{X} be a random variable that symbolises success and is represented by the number six appearing on at least one die.

Now

P(X=0)=P( six does not appear on any of die )=5 / 6 \times 5 / 6=25 / 36

P(X=1)=P( six appears at least once of the die )=(1 / 6 \times 5 / 6)+(5 / 6 \times 1 / 6)=10 / 36=

5 / 18 P(X=2)=P( six does appear on both of die )=1 / 6 \times 1 / 6=1 / 36 Hence, the required probability distribut on is,

\mathrm{E}(\mathrm{X})=\sum_{\mathrm{i}=1}^{\mathrm{n}} \mathrm{x}_{\mathrm{i}} \mathrm{p}_{\mathrm{i}}

Now, we may retrieve the result by substituting the values.

=0 \times \frac{25}{36}+1 \times \frac{5}{18}+2 \times \frac{1}{36}

=\frac{5}{18}+\frac{1}{18}=\frac{6}{18}=\frac{1}{3}

\Rightarrow \mathrm{E}(\mathrm{X})=\frac{1}{3}

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