Probability questions of watch problem

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.

  1. What is the probability that the second watch works?
  2. 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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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
$$P=\frac{78805/80000}{397/400}=\frac{15761}{15880}$$