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13.11 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

(i) $P(E\mid F)$ and $P(F\mid E)$

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A fair die is rolled.

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

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

$E\cap F=\left \{ 3\right \}$

$n(E\cap F)=1$

$n( F)=2$

$n( E)=3$

$P( E)=\frac{3}{6}$ $P( F)=\frac{2}{6}$ and $P(E\cap F)=\frac{1}{6}$

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

$P(E| F)=\frac{\frac{1}{6}}{\frac{2}{6}}$

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

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

$P(F| E)=\frac{\frac{1}{6}}{\frac{3}{6}}$

$P(F| E)=\frac{1}{3}$

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