Solutions - 5040-02

(1) Define events:

Let $D_{4}$ , $D_{6}$ , $D_{12}$ = event of drawing each die.

$P [4 | D_{4}] = \frac{1}{4}$ , $P [4 | D_{6}] = \frac{1}{6}$ , $P [4 | D_{12}] = \frac{1}{12}$ , and $P [D_{4}] = P [D_{6}] = P [D_{12}] = \frac{1}{3}$ .

(2) Apply Bayes’ Theorem:

\begin{matrix} P [D_{6} | 4] = \frac{P [4 | D_{6}] P [D_{6}]}{P [4]} \\ ≫ ≫ \frac{P [4 | D_{6}] P [D_{6}]}{P [4 | D_{4}] P [D_{4}] + P [4 | D_{6}] P [D_{6}] + P [4 | D_{12}] P [D_{12}]} \\ ≫ ≫ \frac{\frac{1}{6} \cdot \frac{1}{3}}{\frac{1}{4} \cdot \frac{1}{3} + \frac{1}{6} \cdot \frac{1}{3} + \frac{1}{12} \cdot \frac{1}{3}} \\ ≫ ≫ \frac{1}{3} \end{matrix}

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Solutions - 5040-02

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