Question

Factories I and II produce phones for brand ABC. I produces 60% of all ABC phones, II produces 40%. 10% of phones produced by I are defective, and 20% of those produced by II are defective. You know that the store where you buy your phones is supplied by one of the factories, but you don't know which one. You buy two phones, and both are defective. What is the probability that the store is supplied by factory II? Make an appropriate assumption of conditional independence.

Answer 1

Given:

$$ \begin{aligned} &P\left(F_{I}\right)=0.6 \\ &P\left(F_{I I}\right)=0.4 \\ &P\left(D \mid F_{I}\right)=0.1 \\ &P\left(D \mid F_{I I}\right)=0.2 \end{aligned} $$

You buy a phone, which is defective. The probability that the store is supplied by factory II is:

$$ \begin{aligned} P\left(F_{I I} \mid D\right) &=\frac{P\left(D \mid F_{I I}\right) \times P\left(F_{I I}\right)}{P\left(D \mid F_{I}\right) \times P\left(F_{I}\right)+P\left(D \mid F_{I I}\right) \times P\left(F_{I I}\right)} \\ &=\frac{0.2 \times 0.4}{0.1 \times 0.6+0.2 \times 0.4} \\ &=0.5714 \end{aligned} $$

You buy two phones, and both are defective. The probability that the store is supplied by factory II is:

$$ 0.5714^{2}=0.3265 $$