Sadly there has been a crime It has been committed by one of

Sadly, there has been a crime. It has been committed by one of two suspects, which we call A and B. Before any investigation is done, we must assume that A and B are equally likely to have committed the crime. After the investigation, it is found that the guilty party had a blood type found in 10% of the population. Suspect A matches this blood type. Sadly, we do not know the blood type of suspect B.

(b) Given the investigation\'s findings, what is the probability that suspect B\'s blood type matches the blood type found on the crime scene? (HINT: you\'ll need to condition on more than one event here).

Solution

Probability of suspect A to commit crime is P(A)

P(A) = 1/2

Probability of suspect B to commit crime is P(B)

P(B) = 1/2

Let E denotes the event when the blood type matches.

P(E/A) = 1/2 * 10/100 = 1/20

P(E/B) = 1/2 * (1-1/10) = 1/2 * 9/10 = 9/20

P(B/E) = P(B)*P(E/B) / (P(A)*P(E/A) + P(B)*P(E/B))

= (1/2 * 9/20) / (1/2 * 1/20 +1/2 * 9/20 )

= 9/40 / 10/40

= 9/10 .

= 9/10 .