Question

A secret government agency has developed a scanner which determines whether a person is a terrorist. The scanner is fairly reliable; 95% of all scanned terrorists are identified as terrorists, and 95% of all upstanding citizens are identified as such. An informant tells the agency that exactly one passenger of 100 aboard an aeroplane in which you are seated is a terrorist. The police haul off the plane the first person for which the scanner tests positive. What is the probability that this person is a terrorist

Answer 1

The probability of a person is a terrorist is
`0.161.`

Explanation:

Let,

`A=` Your neighbor is a terrorist

`B=` Your neighbor tested positive

Now we want to find the value of
`P(A|B)` where

`P(A|B)=(P(B|A)P(A))/(P(B|A)P(A)+P(B|A^c)P(A^c))`

where
`P(A)=\frac 1{100}=1-P(A^c), P(B|A)=0.95,`

`P(B|A^c)=1-0.95=0.05`

Therefore,
`P(A|B)=((0.95)/(100))/((0.95)/(100)+0.05* (99)/(100))`

`=((0.95)/(100))/((0.95+0.05* 99)/(100))=(0.95)/(0.95+4.95)`

`=0.161`

Hence, the probability of a person is a terrorist is
`0.161.`