a You are on a jury and the prosecution tells you that the a

a. You are on a jury and the prosecution tells you that the
accused has the same blood type as evidence found at the crime, and
only one in a thousand people have that type. The defense attorney
points out that the police tested samples from 200 people. If everyone
tested was a random innocent person, what is the probability that a
1/1000 match would have been found by chance?

b. What if there were 10 distinct samples found at the
scene, each matching a separate 1/1000 of the population? That is,
what is the probability that a 1/1000 chance match would have been
found between one of the randomly tested individuals and one of the samples?

Solution

1/1000 being the probability the the blood type matches is very small and we can take Poisson distribution with mean

= 200(1/1000) = 0.2

Thus x follows a Poisson distribution with lemda = 0.2

a) P(X=0) = 0.8187

b) P(x=1 when n =200) = 0.1637