problem 8

2.1.6 problem 8

Let $A_{1}$ be the event that the screen is produced by company $A$ , $B_{1}$ be the event that the screen is produced by company $B$ , and $C_{1}$ be the event that the screen is produced by company $C$ . Let $D$ be the event that the screen is defective.

\begin{matrix} P (A_{1} | D) & = \frac{P (A_{1}) P (D | A_{1})}{P (A_{1}) P (D ∣ ∣ A_{1}) + P (A_{1}^{c}) P (D ∣ ∣ A_{1}^{c})} = \frac{P (A_{1}) P (D | A_{1})}{P (A_{1}) P (D ∣ ∣ A_{1}) + P (A_{1}^{c}) (P (B_{1} ∣ ∣ A_{1}^{c}) P (D ∣ ∣ B_{1}, A_{1}^{c}) + P (C_{1} ∣ ∣ A_{1}^{c}) P (D ∣ ∣ C_{1}, A_{1}^{c}))} = \frac{0.5 * 0.01}{0.5 * 0.01 + 0.5 * (0.6 * 0.02 + 0.4 * 0.03)} = 0.29 \end{matrix}

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