A research study involves $20$% heavy smokers, $30$% light smokers and $50$% non-smokers. Light smokers are twice as likely to die than non-smokers but are half as likely to die than heavy smokers. A person has recently died that was involved in the research study. What is the probability that the person was a heavy smokers?
My answer (is this correct) $P(A) = 0.20$, $P(B) = 0.30$ $P(C) = 0.50$
$P(D|A) = \cfrac47, P(D|B) = \cfrac27, P(D|C) = \cfrac17$
Bayes theorem $$P(A|D) = \frac{P(A)P(D|A)}{P(D)} = \frac{0.20(\frac{4}{7})}{0.20(\frac{4}{7}) + 0.30(\frac{2}{7}) + 0.50(\frac{1}{7})}$$
You answer is correct, but there is a small error in you calculation. You only now the ratio between "likelihood of death" not the exact numbers. Observe the difference.
$P(A) = 0.20$, $P(B) = 0.30$ $P(C) = 0.50$
$P(D|A) = 4a, P(D|B) = 2a, P(D|C) = a$
Bayes theorem $$P(A|D) = \frac{P(A)P(D|A)}{P(D)} = \frac{0.20(4a)}{0.20(4a) + 0.30(2a) + 0.50(a)}$$
It gives the same answer in this case.