What is joint probability?

Joint probability is the probability of two or more events occurring simultaneously.

In probability theory, joint probability is the probability of two or more events occurring simultaneously. It is denoted by P(A and B), where A and B are two events. The joint probability can be calculated using the formula:

P(A and B) = P(A) x P(B|A)

where P(A) is the probability of event A occurring and P(B|A) is the conditional probability of event B occurring given that event A has occurred.

For example, suppose we have two events A and B, where A is the event of rolling a 4 on a fair die and B is the event of rolling an even number on the same die. The probability of event A is 1/6, since there is only one way to roll a 4 out of six possible outcomes. The probability of event B given that event A has occurred is 1/3, since there are two even numbers (2 and 4) out of three possible outcomes (2, 4, and 6) when a 4 is rolled. Therefore, the joint probability of events A and B is:

P(A and B) = P(A) x P(B|A) = (1/6) x (1/3) = 1/18

This means that the probability of rolling a 4 and an even number on a fair die is 1/18.

Joint probability is an important concept in probability theory and is used in many applications, such as in statistical inference, machine learning, and data analysis.