Aresearcher finds that, of 982 men who died in 2002, 221 died from some heart disease. also, of the 982 men, 334 had at least one parent who had some heart disease. of the latter 334 men, 111 died from some heart disease. a man is randomly selected from the group of 982. given that neither of his parents had some heart disease, what is the probability that this man dies of some heart disease?

Within the sample space of 982 men you have two overlapping sets PHHD (ParentHadHeartDesease) with 334 entries => P[PHHD] = 334/982(1), and MDFHD (ManDiedFromHeartDesease) with 221 entries => P[MDFHD] = 221/982(2) with the overlap PHHD*MDFHD (I use * symbol for intersection) with 111 entries => P[PHHD*MDFHD] = 111/982(3) Now: P[MDFHD|NOPHHD] = P[MDFHD*NOPHHD]/P[NOPHHD](4) (NOPHHD = No PHHD). From (1) P[NOPHHD] = 1 - P[PHHD] = 1 - 334/982 = 648/982(5). P[NOPHHD*MDFHD] + P[PHHD*MDFHD] = P[MDFHD] => (from (2) and (3)) P[NOPHHD*MDFHD] = 221/982 - 111/982 = 110/982(6) Finally plugging (5) and (6) into (4) you get: P[MDFHD|NOPHHD] = P[MDFHD*NOPHHD]/P[NOPHHD] = 110/648.