Cracking a lock Suppose you want to pick one of those combo locks you see on bicycles. You have 4 dials you can twirl and you have to guess the right combination. You don't have any other information and so you try randomly twirling the lock. Since there are 10,000 possibilities, the probability that you'll get it right on the first guess is 1/10,000. Not really worth it, considering the bicycle owner is 7 foot 1, with a tattoo saying "Moma". But now suppose you overhear a conversation where you only catch the last digit of the combination, which is 6. Now what's the probability you'll get it right on the first try? You only have to guess three digits now, so that should be 1/1000. Let's see how this works from Bayes's Theorem. Call B the correct combination. Call A all combinations ending with digit 6. What's AB? That's the last digit being 6 intersected with the correct combination. Well the last digit for the correct combination is 6, so $AB = B$. What's P(A), that is the probability of getting a last digit of 6? That's 1/10. So Bayes's Theorem says $P(B\vert A) = P(AB)/P(A) = P(B)/P(A) = (1/10000)/(1/10) = 1/1000$. In this case, it was just easier to solve the problem the first way I did, but in lots of cases, Bayes's Theorem is a lot easier. Now this is obviously not the best way to pick a lock. If you're interested in a life of crime, you're studying the wrong thing. Go become a lawyer.