A DNA sample has been taken at the scene of the crime. Statistically, the probability of an arbitrary individual's DNA profile to coincide with the one taken at the scene is 0.000001 (0.0001%, i.e. one in a million). A DNA test is carried out among 20 suspects, one of which is definitely the culprit. The probability of the test to give the wrong result is 0.00001 (0.001%). Of all 20 tested suspects, only one gets a positive result. What is the probability of him being innocent?
2026-04-03 01:00:34.1775178034
DNA test at scene of crime
47 Views Asked by Bumbble Comm https://math.techqa.club/user/bumbble-comm/detail At
1
Let $X$ be the Bernoulli random variable which value is $1$ if the test is positive on the culprit, and let $Y$ be the random number of positive tests on the 19 innocent people ($Y$ obeys a binomial law and is independent from $X$).
The probability that you are looking for is \begin{equation} P(X = 0 | X + Y = 1) = \frac{P(X = 0 \text{ and } X + Y = 1)}{P(X + Y = 1)} = \frac{P(X=0) P(Y=1)}{P(X=0)P(Y=1) + P(X=1)P(Y=0)} \end{equation} The rest should be easy.