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.
  
  
*
  
*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.

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