Conditional probability

Conditional probability is the probability of an event computed by taking into account the fact that another event has already occurred. The new information restricts the universe of possible cases: we no longer reason over all the initial outcomes, but only over those that remain compatible with what we know.

The mental image is that of a spotlight narrowing the field. Before the information, we look at the whole array of possibilities; the information switches off the zones now excluded, and we recompute the proportions inside the zone still lit.

The calculation relates the favorable cases that survive the information to the number of cases that remain possible. In a deck of 52 cards, 26 are red and 13 are hearts. Drawn at random, a card has one chance in four, that is 13/52, of being a heart. But if we know it is red, the universe shrinks to the 26 red cards, among which 13 are hearts: the probability rises to 13/26, that is one chance in two.

We must not confuse "probability of A given B" with "probability of B given A," which are generally different. The probability that a card is red given that it is a heart equals 1, since all hearts are red, whereas the reverse is only 1/2. Confusing the two is a common reasoning error.

When the information changes nothing about the probability, the two events are said to be independent. Bayes' theorem, for its part, formalizes how to reverse the conditioning in order to update a probability in light of an observed clue.

This concept sheds light on the site's draws without replacement: removing a name already drawn changes the chances of the following ones, because each removal conditions the state of the hat for the next round.