Joint Probability: Definition, Formula, and Example

Joint Probability

Investopedia / Jiaqi Zhou

What Is a Joint Probability?

Joint probability is a statistical measure that calculates the likelihood of two events occurring together and at the same point in time. Joint probability is the probability of event Y occurring at the same time that event X occurs.

The Formula for Joint Probability Is

Notation for joint probability can take a few different forms. The following formula represents the probability of events intersection:

 P   ( X Y ) where: X , Y = Two different events that intersect P ( X  and  Y ) , P ( X Y ) = The joint probability of X and Y \begin{aligned} & P\ \left ( X\bigcap Y \right ) \\ &\textbf{where:}\\ &X, Y = \text{Two different events that intersect}\\ &P(X \text{ and } Y), P(XY) = \text{The joint probability of X and Y}\\ \end{aligned} P (XY)where:X,Y=Two different events that intersectP(X and Y),P(XY)=The joint probability of X and Y

What Does Joint Probability Tell You?

Probability is a field closely related to statistics that deals with the likelihood of an event or phenomena occurring. It is quantified as a number between 0 and 1 inclusive, where 0 indicates an impossible chance of occurrence and 1 denotes the certain outcome of an event.

For example, the probability of drawing a red card from a deck of cards is 1/2 = 0.5. This means that there is an equal chance of drawing a red and drawing a black; since there are 52 cards in a deck, of which 26 are red and 26 are black, there is a 50-50 probability of drawing a red card versus a black card.

Joint probability is a measure of two events happening at the same time, and can only be applied to situations where more than one observation can occur at the same time. For example, from a deck of 52 cards, the joint probability of picking up a card that is both red and 6 is P(6 ∩ red) = 2/52 = 1/26, since a deck of cards has two red sixes—the six of hearts and the six of diamonds. Because the events "6" and "red" are independent in this example, you can also use the following formula to calculate the joint probability:

 P ( 6 r e d ) = P ( 6 ) × P ( r e d ) = 4 / 5 2 × 2 6 / 5 2 = 1 / 2 6 P(6 \cap red) = P(6) \times P(red) = 4/52 \times 26/52 = 1/26 P(6red)=P(6)×P(red)=4/52×26/52=1/26

The symbol “∩” in a joint probability is referred to as an intersection. The probability of event X and event Y happening is the same thing as the point where X and Y intersect. Therefore, joint probability is also called the intersection of two or more events. A Venn diagram is perhaps the best visual tool to explain an intersection:

Image by Julie Bang © Investopedia 2019

From the Venn above, the point where both circles overlap is the intersection, which has two observations: the six of hearts and the six of diamonds.

The Difference Between Joint Probability and Conditional Probability

Joint probability should not be confused with conditional probability, which is the probability that one event will happen given that another action or event happens. The conditional probability formula is as follows:

 P ( X , g i v e n   Y )  or  P ( X Y ) P(X, given~Y) \text{ or } P(X | Y) P(X,given Y) or P(XY)

This is to say that the chance of one event happening is conditional on another event happening. For example, from a deck of cards, the probability that you get a six, given that you drew a red card is P(6│red) = 2/26 = 1/13, since there are two sixes out of 26 red cards.

Joint probability only factors the likelihood of both events occurring. Conditional probability can be used to calculate joint probability, as seen in this formula:

 P ( X Y ) = P ( X Y ) × P ( Y ) P(X \cap Y) = P(X|Y) \times P(Y) P(XY)=P(XY)×P(Y)

The probability that A and B occurs is the probability of X occurring, given that Y occurs multiplied by the probability that Y occurs. Given this formula, the probability of drawing a 6 and a red at the same time will be as follows:

 P ( 6 r e d ) = P ( 6 r e d ) × P ( r e d ) = 1 / 1 3 × 2 6 / 5 2 = 1 / 1 3 × 1 / 2 = 1 / 2 6 \begin{aligned} &P(6 \cap red) = P(6|red) \times P(red) = \\ &1/13 \times 26/52 = 1/13 \times 1/2 = 1/26\\ \end{aligned} P(6red)=P(6red)×P(red)=1/13×26/52=1/13×1/2=1/26

Statisticians and analysts use joint probability as a tool when two or more observable events can occur simultaneously. For instance, joint probability can be used to estimate the likelihood of a drop in the Dow Jones Industrial Average (DJIA) accompanied by a drop in Microsoft’s share price, or the chance that the value of oil rises at the same time the U.S. dollar weakens.