Criminal investigation

frenchX · January 9th, 2011, 9:02 pm

A little brainteaser with conditionnal probability A FBI special agent is in charge of a criminal case and at this stage of the investigation he is convinced at 60% that his main suspect is guilty. Another clue is then discovered: the criminal is left handed. We know that 7% of the population is left handed and the main suspect is in this case too. What is the new probability he is the criminal now ?

DevonFangs · January 10th, 2011, 10:23 am

I think it is 0.8599.Let's say P(G) = prob he's guilty, P(NG) = prob he's not guilty, P(L) = prob he's left-handed and P(NL) = prob he's not left handed.Then according to the detective you have P(G) = 0.6, and you know P(L) = 0.07 and P(L|G) = 1. Then using conditional probability,P(G|L) / P(NG|L) = P(G) P(L|G) / ( P(NG) P(L|NG) ) = P(G) P(L|G) / P(L) = 0.6/0.07 = 8.571Then using P(G|L)+P(NG|L)=1, you can get the answer.

DevonFangs · January 10th, 2011, 12:21 pm

Which is obviously meaningless and wrong, so I will think about that again

DevonFangs · January 10th, 2011, 12:49 pm

OK it was simple (I forgot to divide by 0.4). Or ever simpler,P(G|L) = P(G) P(L|G) / (P(G) P(L|G) + P(NG) P(L|NG) ) = 0.9554

BFP · January 10th, 2011, 1:22 pm

Vanubis · January 18th, 2011, 2:35 pm

Maybe another way to answer to this question.Before the clue of left-hander was discovered, we have:60% of being guilty for the main suspect and 40% for anybody else (as we suppose someone is guilty).With the clue, no change for our main suspect (still 60%) and now 2.8%= 40% * 7% for the rest of the world (as we know nothing about it).So we find the same result 60%/(60%+2.8%).