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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 ?

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.

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

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

P(G)=0.6 P(L)=0.07 P(L|G)=1P(G|L) = P(G).P(L|G)/(P(G).(P(L|G)+P(NG).P(L|NG)P(L|NG) = P(L and NG)/NG = (0.4*0.07)/0.4=0.07=0.6/(0.6+0.4*0.07)=95.54%

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%).