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Factory A produces 1 bad watch in 100 and factory B produces 1 bad watch in 200. You are given two watches from one of the factories (chosen with equal probability) and you don’t know which one.

- What is the probability that the second watch works?
- Given that the first watch works, what is the probability that the second watch works?

According to me, the answer should be $\frac{397}{800}$ for both.

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- Mapping CDF's to each other

For part 1, since the watches are independently picked:

$$P=\frac12\cdot\frac{99}{100}+\frac12\cdot\frac{199}{200}=\frac{397}{400}$$

which is not $\frac{397}{800}$.

For part 2, use the law of total probability. The probability that both watches work is

$$\frac12\cdot\frac{99^2}{100^2}+\frac12\cdot\frac{199^2}{200^2}=\frac{78805}{80000}$$

The probability of the first watch working is the same as the answer to part 1. Therefore

$$P=\frac{78805/80000}{397/400}=\frac{15761}{15880}$$

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