Joint Probability: Definition, Formula, and Example

What Is Joint Probability?

A joint probability is the chance that two or more events will happen at the same time. For a joint probability to work, both events must be independent of one another. For instance, it's the likelihood of flipping a coin and getting heads and rolling a die and getting a six. Another example is rolling two dice and both landing on a three. You can visualize joint probabilities using Venn diagrams. Joint probabilities help statisticians, data analysts, and financial professionals draft models, assess risk, and make investment decisions.

Formula and Calculation of Joint Probability

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

 $\begin{aligned} & P\ \left ( X\bigcap Y \right ) \\ &\textbf{where:}\\ &X, Y = \text{Two different events that intersect}\\ &P(X \text{ and } Y), P(XY) = \text{The joint probability of X and Y}\\ \end{aligned}$​P (X⋂Y)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 phenomenon occurring. It is quantified as a number between 0 and 1, 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 there is an equal chance of drawing a red and black card since there are 26 of each in a deck. As such, there is a 50-50 probability of drawing a red card versus a black card.

Joint probability measures two events that happen at the same time. It can only be applied to situations where more than one observation can occur at the same time. So the joint probability of picking a card that is both red and 6 from a deck 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 red and 6 are independent, you can also use the following formula to calculate the joint probability:

$P(6 \cap red) = P(6) \times P(red) = 4/52 \times 26/52 = 1/26$P(6∩red)=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, the 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:

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.

Joint Probability vs. 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:

$P(X, given~Y) \text{ or } P(X | Y)$P(X,given Y) or P(X∣Y)

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

$P(X \cap Y) = P(X|Y) \times P(Y)$P(X∩Y)=P(X∣Y)×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:

$\begin{aligned} &P(6 \cap red) = P(6|red) \times P(red) = \\ &1/13 \times 26/52 = 1/13 \times 1/2 = 1/26\\ \end{aligned}$​P(6∩red)=P(6∣red)×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.

Example of Joint Probability

Let's highlight another example to show how joint probability works. This example uses dice and we want to find out what the probability is that you'll roll a four on each die when you roll them. Remember, there are six sides to each one.

In order to determine the joint probability, we first need to determine the probability of each roll:

• The chance of rolling a three on the first die is 1/6
• The chance of rolling a three on the second die is 1/6
  

Now we can use the joint probability formula noted above to figure out what the joint probability is for this event by multiplying each individual event together.

1/6 x 1/6 = 1/36

This means that there is a 1/36 chance of rolling two fours using a pair of dice.

The Bottom Line

Probability refers to the likelihood that an event will take place. But when two variables are involved, you may have joint probability. This is a statistical measure that can tell you whether two independent events are likely to occur at the same time. It is an important metric for statisticians who use it to determine relationships between two sets of variables, such as the returns of two different companies or high winds and rainfall in weather forecasting. But one thing it doesn't indicate, though, is how the two influence each other.