Probability of finger print

Question

A person is a suspect in a criminal investigation, and there is a 25% chance they are guilty (event G). If they are guilty, there is a 90% chance their fingerprints would be found at the crime scene. If they are not guilty, there is a 10% chance their fingerprints would be found.

Suppose in addition to fingerprints, DNA evidence is found (event D). The prob-ability of leaving DNA evidence is also 10% if they are not guilty and 90% if they are guilty. Leaving evidence of different types is independent, conditional on their guilt/innocence. What is the probability that the person is guilty, given BOTH fingerprints and DNA evidence are found?

We want $P(G|DF) = P(GDF)/P(DF)$, we know $P(D | \overline{G}) = 0.1, P(D | G ) = 0.9$

Can I get a hint?

Answer 1

For any events $a$ and $b$, $$P(a|b) = P(b|a)P(a)/P(b)$$ $$P(b) = P(b|a)P(a) + P(b|\bar a)P(\bar a)$$ therefore $$P(a|b) = {P(b|a)P(a) \over P(b|a)P(a) + P(b|\bar a)P(\bar a)}$$

$$P(G|DF) = {P(DF|G)P(G) \over P(DF|G)P(G) + P(DF|\bar G)P(\bar G)}$$ $$P(G|DF) = {0.9^2\cdot 0.25 \over 0.9^2 \cdot 0.25 + 0.1^2 \cdot 0.75} = {27 \over 28} $$