Conditional probability is a measure of the likelihood of an event that is conditioned on another event, usually referred to as the conditioning event. It is a type of probability, closely related to marginal and joint probability, but with an additional condition that must be met for the probability to be meaningful. It is the concept of evaluating the probability of one event occurring given that another event has already occurred, and can be expressed mathematically as P(B|A), where A and B represent two events.

Conditional probability is closely linked to Bayes’ Theorem, a mathematical formula used to calculate conditional probabilities. Bayes’ Theorem is based on the premise that, given some information or evidence, the probability of an event occurring should be re-evaluated. To illustrate this concept, suppose you have a box with 1 red ball and 9 blue balls. If you randomly select one ball from the box, the unconditional probability of drawing a red ball is 1 in 10 (or 10%). That is, the unconditional probability of drawing a red ball is 10% regardless of which ball is chosen.

However, if the box is shaken and the ball is returned to it, now the chances of drawing a red ball from the box has changed. This is conditional probability, and Bayes’ Theorem can be used to calculate the new probability of drawing a red ball in this situation. Using Bayes’ Theorem, the probability of drawing a red ball, assuming the ball was seen in the box, is now 1/2. That is, the conditional probability of drawing a red ball is now 50% if the ball was observed previously within the box.

In summary, conditional probability measures the chances that some outcome occurs given that another event has also occurred. It can be expressed as the probability of event B given event A, written as P(B|A). Bayes' theorem is a mathematical formula used in calculating conditional probability, taking into account the prior evidence or information of the occurrence of event B, given the occurrence of event A.