How does conditional probability revise an estimate using background knowledge?

"I mentioned that Bayesians are especially concerned with the idea of revising estimates about probability based on new information that may come to light. This notion can be crystallized in the idea of conditional probability. When we talk about the conditional probability of an event A, we mean “what’s the probability that A occurs, given that I know some other event K has also occurred?” Think of K as “background knowledge”: it’s additional information which, when known, may influence how likely we think A is to have occurred. It can be mathematically computed as follows: Pr(A | K) = Pr(A ∩ K) / Pr(K). We pronounce Pr(A | K) as “the probability of A given K.” It is the conditional probability of A, or “the probability of A conditioned on K.” We’ll sometimes call plain old Pr(A) the a priori probability, or the prior probability if we don’t want to sound Latin. The prior is simply the original unadjusted probability, if we aren’t privy to the background information K.\n\nLet’s go back to American Idol. We know that the probability of an underage winner is only .4, because U = {Kelly, Fantasia}, and we estimate that each of them has a .2 probability of winning. So it seems more likely than not that our winner will be over 21. But wait: suppose we had some additional information. Just before the outcome is announced, news is leaked through a Rupert Murdoch news source that the winner is a woman! If we believe this reporter, does that change our expectation about how old the winner is likely to be? Indeed it does. Knowing that the winner is female eliminates Dave from consideration. Looking back at Figure 4.1, we can see that once we know Dave is out of the running, the remaining pool consists of just F, which includes Kelly, Fantasia, and Carrie. The question is, how do we update our probability from .4 to reflect the fact that only these three ladies are left? In this case F is the background knowledge: we know that the event F has occurred. And we want to know how likely U is to also have occurred. This is found easily: Pr(U | F) = Pr(U ∩ F) / Pr(F) = Pr({Kelly, Fantasia}) / Pr({Kelly, Fantasia, Carrie}) = .4 / .5 = .8. Our estimated chance of an underage winner doubled once we found out she was female (even though we don’t yet know which female). If you stare at the equation and diagram, you’ll see the rationale for this formula. Kelly and Fantasia originally had only .4 of the entire probability between them. But once David was axed, the question became: “what percentage of the remaining probability do Kelly and Fantasia have?” The answer was no longer .4 out of 1, but .4 out of .5, since only .5 of the whole was left post-David. This is why we divided by Pr(F): that’s what we know remains given our background fact.\n\nNow in this case, the conditional probability was higher than the original probability. Could it ever be lower? Easily. Consider the probability of a rock-star winner, Pr(R). A priori, it’s .7. But again, let’s say we had information leaked to us that the winner, whoever she may be, is female. We can now update our estimate: Pr(R | F) = Pr(R ∩ F) / Pr(F) = Pr({Kelly}) / Pr({Kelly, Fantasia, Carrie}) = .2 / .5 = .4. You see, once we find out that David is no longer a possibility, our only remaining hope for a rock star is Kelly. And she has only 40% of the probability that’s left over. Note that this is a higher chance for her personally — she’s got to be excited by the press leak — but it’s lower for rock stars, of which she is only one (and evidently, not the predicted strongest)."