A lesson in conditional probability

Probably the hardest notion you are likely to encounter in basic statistics class is conditional probability. It can be very counter-intuitive, and requires real care to get right.

It's easiest to understand by context. Consider that an AIDS test might be 97% accurate in the sense that it only generates false negatives 3% of the time. The same test might be 99.5% accurate because it only gives false positives 0.5% of the time.

If you, an ordinary person, with no special risk factors test positive, what is the probability you have the disease? Maybe you think 99.5% likely?

Not particularly likely it turns out. See, among people without any particular risk factors, AIDS is pretty rare. Of 1000 people who got tested, you might only expect 1 to actually be sick, but you'd get 5 false positives in a group that size.

So a single positive test really only means your probability of being sick is like 1 in 6. It's a lot less than you think, and it's not that easy to wrap your brain around. It's also the reason they retest people who test positive.

It's called conditional probability, because the question is about the probability of something being true based on information you already know (the condition). You are asking what is the probability of being sick, given the condition of a positive test.

Conditional luck

The reason for the statistics lesson is that Linda and I have been talking about conditional luck lately. People are always telling her that she is lucky.

Finally, I told her that she is conditionally lucky. Given that she had a very bad stroke, she has had a remarkably good and improbable amount of recovery. (In my way of thinking good*improbable=lucky.) That's what people see and what they can wrap their brains around.

It's not quite the same thing as being lucky. But it's better than being pure unlucky.