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Joint probability is the likelihood of two or more events happening together. The multiplication rule is used to calculate independent and dependent events. Independent events use P(A) x P(B), while dependent events use P(A) x P(B|A). Substitution can also affect the calculation.
Joint probability (P) refers to the probability of two events occurring simultaneously, where an event can be understood as anything being measured, such as the drawing of a specific card or the roll of a die. Typically, the term joint means two simultaneous occurrences, but sometimes it can be applied to more than two events. There are specific rules in statistics and probability that govern how to evaluate this probability. The simplest methods use special multiplication rules. Also, independent events or the use of substitution require consideration and modification calculations.
The simplest form of joint probability occurs when two independent events are considered. This means that the outcome of each event does not depend on the other. For example, in rolling two dice, an individual might want to know the joint probability of rolling two sixes in a single roll. Each event is independent, and rolling a six against one die does not affect what happens with the second one.
The multiplication rule here is that the probability of A and B or P(A and B) equals the probability of P(A) times 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 the joint probability for dependent events is evaluated, the multiplication rule changes. While these events are “joint,” one influences the outcome of the other. These changes must be considered when making a calculation.
Consider drawing two red cards from a regular 52-card deck. Since half of the cards are red, the probability of drawing a red card or P(A) is 1/2. Even though the cards are drawn at the same time, the second event has a different level of probability as there are now 51 cards and 25 red. P(B), drawing a second red card, is actually 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 correct multiplication rule to use, it is important to consider the concept of substitution. If the first red card has been drawn and a new red card has been placed into the stock before the second card has been drawn, these two events become independent. The joint probability with substitution acts as a simple independent probability and is evaluated as P(A) × P(B).
