How does conditional probability describe the chance of an event after another event has occurred?

In mathematics, conditional probability describes the likelihood of an event occurring given the knowledge that another event has already taken place. It is essential for analyzing dependent events, where the outcome of one event influences the probability of another.12

The probability of an event A occurring given that event B has occurred is denoted as $P(A|B)$. Assuming the probability of B is greater than zero, it is calculated using the following formula:1

$$P(A|B)=\frac{P(A\cap B)}{P(B)}$$

In this equation, $P(A\cap B)$ represents the probability of both events occurring (their intersection), and $P(B)$ is the probability of the conditioning event.1

When events are independent, the occurrence of one does not change the probability of the other, meaning conditional probability is not required. However, for dependent events—such as drawing cards from a deck without replacement—the odds shift based on previous outcomes. To calculate "reverse" probabilities, such as the likelihood of an event having occurred given a later dependent result, mathematicians use Bayes's theorem.123