Question

4. There are three equally likely suspects for a crime. Suspect A is a short man, Suspect B is a tall man, and Suspect C is a tall woman. Only one of them was at the crime scene. Two witnesses report seeing a short person at the crime scene, and one witness reports seeing a woman at the crime scene. In your experience, witness reports are 90% reliable (each witness, independently, is 90% likely to make a correct report, and 10% likely to make a false report). Based only on the testimony of these three witnesses, what are the probabilities of each suspect being guilty

Answer 1

The probabilities of each suspect being guilty are

`P(S)  =  0.243`

`P(W)  =  0.027`

`P(T) = 0.001`

Explanation:

Generally from the question we are told that

The number of suspects is N = 3

The number of witnesses that reported seeing a short man is k = 2

The number of witnesses that reported seeing a woman is u = 1

The probability of a witness being correct is p = 0.90

The probability of a witness not being correct is q 0.10

Generally the number of witnesses that reported seeing a tall man is h = 0

Generally the probability the short man being guilty is mathematically represented as

`P(S) = ^N C_k * p^k * q^(N - k )`

Here C stands for combination

`P(S) = ^3 C_2 * (0.90)^2 * (0.10)^(3 - 2 )`

`P(S)  =  0.243`

Generally the probability the woman being guilty is mathematically represented as

`P(W)  =  ^N C_u  *  p^u  *  q^(N - u )`

=>
`P(W)  =  ^3 C_1  *  p^1  *  q^(3 - 1 )`

=>
`P(W)  =  0.027`

Generally the probability the tall man being guilty is mathematically represented as

`P(T)  =  ^N C_h  *  p^h  *  q^(N - h )`

=>
`P(T)  =  ^3 C_0  *  p^0  *  q^(3 - 0 )`

=>
`P(T)  = 0.001`