conditional probability

Conditional probability answers the question: now that I know one thing has happened, how does that change the chance of another? Before you learn anything, the chance a stranger will die this year might be tiny. But once you learn they are 90 years old, that chance jumps. The new, updated probability — computed in the light of the information you now have — is a conditional probability.

We write the probability of A given that B has happened as P(A | B), and it is defined as P(A and B) divided by P(B). In words: out of all the ways B can happen, what fraction also have A happening? Suppose 1% of people have a certain illness and the test is positive in 0.9% of all people who both have it and test positive; then among the people who test positive, the chance they truly have it is that overlap divided by the chance of a positive test. Conditioning effectively shrinks the sample space down to just the part where B is true and re-measures A inside it.

Conditioning is the everyday breathing of actuarial work. A mortality rate q_x is really a conditional probability: the chance of dying within the year given you have survived to exact age x. Premiums for a substandard life, the chance of a large claim given a claim has occurred, the survival of a policy given it lasted ten years — all are conditional. Updating probabilities as new information arrives is exactly what underwriting and reserving do.