Joint probability (P) refers to the likelihood of two events occurring at the same time, where an event can be understood as anything being measured, like a specific card being drawn or a roll of the dice. Typically, the term joint means two simultaneous occurrences, but it may sometimes be applied to more than two events. There are specific rules in statistics and probability that govern how to assess this likelihood. The simplest methods use special multiplication rules. Additionally, independent events or the use of replacement require consideration and change calculations.
The simplest form of joint probability occurs when two independent events are considered. This means the outcome of each event doesn't depend on the other. For example, in rolling two dice, an individual might want to know the joint probability of getting two sixes in a single roll. Each event is independent, and getting a six on one die doesn't influence what happens with the second.
The multiplication rule in this instance is that probability of A and B or P(A and B) is equal to the probability of P(A) multiplied by P(B). This can also be expressed as P(A × B). There is a 1/6 chance of rolling a six on a six-sided die. So P (A and B) is 1/6 × 1/6 or 1/36.
When joint probability is evaluated for dependent events, the multiplication rule changes. Though such events are "joint," one influences the outcome of other. These changes must be considered when making a calculation.
Consider the possibility of drawing two red cards from a normal 52-card deck. Since half the cards are red, the probability of taking out one red card or P(A) is 1/2. Even if the cards are simultaneously drawn, the second event has a different probability level as there are now 51 cards and 25 red ones. P(B), drawing a second red card, is really P (B | A), which reads as B given A. This is 25/51, instead of 1/2.
The formal multiplication rule for dependent events is P(A) × P(B | A). For this example, the joint probability of two red cards is 1/2 × 25/51. This equals 25/102 or, as is more common, can be written as a decimal with three places: 0.245.
When determining the right multiplication rule to use, it is important to consider the concept of replacement. If the first red card was drawn and a new red card was placed in the deck prior to drawing the second card, these two events become independent. Joint probability with replacement acts like simple independent probability, and is evaluated as P(A) × P(B).