Bayes' theorem conditional probability
Probability is not a fixed number. When one new piece of information arrives, the probability has to be computed again at once. Even a 30% chance of rain changes the moment you see a dark cloud. Conditional probability and Bayes' theorem are the arithmetic of how a probability changes once a condition is attached. This lesson starts from the intuition of narrowing the sample space, goes on to Bayes updating a belief, and follows with the eye the famous paradox of a test for a rare disease. A condition rewrites the number.
Conditional probability is the probability after you know something. Once you know that one event has happened, the sample space narrows to that event, and the share the other event takes inside it is the new probability. So P(A|B) = P(A∩B)/P(B). Turn the condition on and the denominator shrinks, and you see how the probability changes. Knowing narrows the space.
Bayes' theorem is the tool that turns a conditional probability around. Multiply how much you believed the hypothesis first by how well the evidence fits that hypothesis, and you get the belief after seeing the evidence. In symbols, P(H|E) = P(E|H)P(H)/P(E). Push the first belief and the fit of the evidence, and the updating of the belief follows as it stands. Turning it around repairs the belief.
Even a very accurate test falls into a trap when the disease is rare. Healthy people overwhelmingly outnumber the sick, so even if only a tiny fraction of them are falsely positive, that number exceeds the true patients. So even when the test is positive, the probability of actually having the disease stays low. This is the classic example where Bayes' theorem shows itself plainly. A rare event fools the test.
A probability tree draws, as branches, a process in which events happen in stages. Multiply the probabilities of the branches along one path and you get the joint probability of that ending. The second fork is conditional, so its value changes with the result of the earlier stage. Add the probabilities of every leaf and the sum is always exactly 1. The sum of the leaves fills the whole.
Two events are independent when knowing that one has happened does not change the probability of the other at all. Only then does P(A∩B) = P(A)P(B) hold. If one is tied to the other, this multiplication breaks. In area, you can see at a glance whether the real A∩B block matches the dotted rectangle predicted by the product, or misses it. Only when the product matches are they independent.