# Conditional Probability & Bayes' Theorem

[Problem]
Stores A, B, and C have 50, 75, and 100 employees, respectively, 50, 60, and 70 percent of these are women. Resignations are equally likely among all employees, regardless of sex. One employee resigns and this is a woman. What is the prob. she works in store C?

[Solution]
Store A: 25F
Store B: 45F
Store C: 70F

P(W|A) = 25/50 (Prob. it was a woman resign given store A)
P(W|B) = 45/75
P(W|C) = 70/100

P(A) = P(B) = P(C) = 1/3 ?

or

P(A) = 50/225
P(B) = 75/225
P(C) = 100/225 ?

P(C|W) = $$\frac{P(W|C)P(C)}{P(W|C)P(C) + P(W|A)P(A) + P(W|B)P(B)}$$

Does this look right?

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EnumaElish
P(C|W) = P(C and W)/P(W) = P(W|C)P(C)/$\sum_s$P(W|s)P(s), so it's correct.