I am practicing for my Probability exam, and here's a question that I am unable to solve.
The question is from Introduction to Probability by Joseph K. Blitzstein
Jessica Hwang
Question says that it is 23 times more likely for a smoker to develop lung cancer than a non-smoker, and 21.6% men in U.S smoke. We are required to find probability that a person is a smoker, if he develops cancer.
I believe the question is related to Conditional probability but I am not sure how to right 23 times more likely in terms of probability. Any help will be appreciated. Thank you!
So far I have defined two events:
A: Man has cancer.
B: Man Smokes.
We have P(B), and I believe we need to find P(B|A), I am stuck and don't know how to move forward.
Thank you!
Best Answer
Let $P(A)$ be the probability that a man has cancer. You can partition this probability over other events. There are 2 such partitions - a man smokes and a man does not smoke.
$$P(A | \text{ man smokes }) = 23P(A| \text{ man does not smoke})$$
Also notice that $P(A) = P(A | \text{ man smokes }).P( \text{ man smokes }) + P(A| \text{ man does not smoke}).P( \text{ man does not smoke })$
You are given that $P(\text{ man smokes }) = 0.216$. Can you now apply Bayes theorem?