Solutions - 5030-05

(1) Apply the Law of Total Probability to find $P[B]$:

$$\begin{array}{c}P[B]=P[B|A_1]P[A_1]+P[B|A_2]P[A_2]+P[B|A_3]P[A_3]\\ \\ \gg \gg 0.5\cdot 0.3+0.5\cdot 0.4+0.6\cdot 0.3\gg \gg 0.53\end{array}$$

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(2) Apply Bayes’ Theorem to find $P[A_1|B]$:

$$\begin{array}{c}P[A_1|B]=\frac{P[B|A_1]P[A_1]}{P[B]}\\ \\ \gg \gg \frac{0.5\cdot 0.3}{0.53}\gg \gg \approx 0.2830\end{array}$$

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(3) Apply Bayes’ Theorem to find $P[A_2|B]$:

$$\begin{array}{c}P[A_2|B]=\frac{P[B|A_2]P[A_2]}{P[B]}\\ \\ \gg \gg \frac{0.5\cdot 0.4}{0.53}\gg \gg \approx 0.3774\end{array}$$

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(4) Apply Bayes’ Theorem to find $P[A_3|B]$:

$$\begin{array}{c}P[A_3|B]=\frac{P[B|A_3]P[A_3]}{P[B]}\\ \\ \gg \gg \frac{0.6\cdot 0.3}{0.53}\gg \gg \approx 0.3396\end{array}$$