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11.11 A fair die is rolled. Consider events $E=\left \{ 1,3,5 \right \},F=\left \{ 2,3 \right \}$ and $G=\left \{ 2,3,4,5 \right \}$ Find

(ii) $P(E\mid G)$ and $P(G\mid E)$

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A fair die is rolled.

Total outcomes $=\left \{ 1,2,3,4,5,6 \right \}=6$

$E=\left \{ 1,3,5 \right \}$ , $G=\left \{ 2,3,4,5 \right \}$

$E\cap G=\left \{ 3,5\right \}$

$n(E\cap G)=2$

$n( G)=4$

$n( E)=3$

$P( E)=\frac{3}{6}$ $P( G)=\frac{4}{6}$ $P(E\cap F)=\frac{2}{6}$

$P(E| G)=\frac{P(E\cap G)}{P(G)}$

$P(E| G)=\frac{\frac{2}{6}}{\frac{4}{6}}$

$P(E| G)=\frac{2}{4}=\frac{1}{2}$

$P(G| E)=\frac{P(G\cap E)}{P(E)}$

$P(G| E)=\frac{\frac{2}{6}}{\frac{3}{6}}$

$P(G| E)=\frac{2}{3}$

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