A certain blood type matches one person in 5 000. A man stands accused of a
serious crime, and you, a member of the jury in his trial, have estimated
there’s a 10% chance that he’s guilty on the evidence so far. You then hear
that his blood type matches that of some found at the scene. What’s your new
estimate of the probability of his guilt based on this new evidence? Assume
that the blood found at the scene is definitely that of the perpetrator.
I've figured out the likelihood ratio, but for the posterior odds of guilt I would need some measure of population. How should I solve this?
Best Answer
For purposes of this problem, there are two ways the rare blood could have appeared on the crime scene: either the defendant is guilty or he is innocent and the guilty party had the same blood type. Thus the probability that the rare type would be observed is $$.1\times 1 +.9\times \frac 1{5000}$$
Since the defendant's guilt accounts for the $.1\times 1$ term, the revised estimate for his guilt is $$\frac {.1\times 1}{.1\times 1 +.9\times \frac 1{5000}}=\boxed {.9982}$$
To be sure: as is always the case in problems like this, we are making a lot of assumptions. These computations are not so clean in the real world. The guilty party may have left the rare blood in an attempt to incriminate the defendant. The police may have arrested the man because of the blood, so your prior estimate may have already used that evidence. Blood type might be correlated with the killing in some way (maybe several members of the defendant's family have that rare type). And so on.