A Condition Changes the Probability
A probability is not a fixed number. The moment one new piece of information arrives, it must be recomputed. Even a 30% chance of rain changes the instant you see dark clouds gather. Conditional probability and Bayes' theorem are exactly the arithmetic of how a probability shifts once a condition is attached. Here we start from the intuition of shrinking the sample space, move to Bayes updating a belief, and reach the famous paradox of testing for a rare disease, all visually.
Conditional probability is the probability after you learn something. Once you know event B happened, the sample space shrinks to B, and the share of A inside it becomes the new probability. That is why P(A|B) = P(A and B)/P(B). Toggle B on the grid and watch the denominator shrink while the probability changes.
Bayes' theorem is the tool for flipping a conditional around. Multiply how much you believed the hypothesis to begin with (the prior) by how well the evidence fits it (the likelihood), and you get your belief after seeing the evidence (the posterior). That is P(H|E) = P(E|H)P(H)/P(E). Slide the prior and the likelihood to watch the belief update.
Even a very accurate test falls into a trap when the disease is rare. Healthy people vastly outnumber the sick, so even a tiny fraction of them testing positive by mistake can outnumber the true patients. As a result, the chance of really being sick given a positive result stays low. This is the classic case where Bayes' theorem shows its teeth.
A probability tree draws a step-by-step process as branches. Multiply the branch probabilities along one path and you get the joint probability of that ending. The second split is conditional, so its values depend on the earlier outcome. Add the probabilities of all the leaves and they always sum to exactly 1.
Two events are independent when knowing one happened does not change the probability of the other at all. Only then does P(A and B) = P(A)P(B) hold. If they are dependent, that multiplication breaks. Seen as areas, you can judge it at a glance by whether the actual A-and-B block matches or misses the dashed rectangle predicted by the product.