Find the probability that the individual is 65 or older and suffering from mild heart failure?

Question

A study indicates 10% of the population is 65 years old or older and that 1% of the total population is afflicted with mild heart failure. Furthermore, 10.4% of the population age 65 older or suffers from mild heart failure.

Answer 1

If the population is (arbitrarily) 10,000 people, then 10% of 10,000=1000 over 65s; and 1% of 10,000=1000 with mild heart failure. 

Consider 4 groups of people: A, B, C, D.

Let A+B=1000 be the over 65s (65⁺), and B+C=100 be those with heart failure (HF). (D are under 65 and don't have heart failure. A+B+C+D=10000.) B are 65⁺ + HF=10.4% of (A+B)=104.

The number of people who are 65⁺ or HF, but not both=(A+B)+(B+C)-B=A+B+C=1000+100-104=996.

996/10000=0.0996 or 9.96%, the probability of 65⁺ or HF. (D=10000-996=9004, under 65 and no HF.)