Joint Probability: Definition, Formula & Examples

What is Joint Probability?

Joint probability is the likelihood that two or more events will coincide. Knowing how to calculate them allows you to solve problems such as the following. What is the probability of:

Getting two heads in two coin tosses?
Consecutively drawing two aces from a deck of cards?
The next customer being a woman who buys a Mac computer?
A bike rental customer getting both a flat front tire and a flat rear tire?

Statisticians use the notation of P(A ∩ B) to indicate the joint probability of events “A” and “B” occurring together. For example, P(F ∩ Mac) denotes the likelihood of a female buying a Mac. Equivalent variants of this notation include P(A and B) and P (A,B).

The notation includes the symbol “∩,” which signifies an intersection. This intersection specifies how two or more events, such as A and B, coincide. Consequently, joint probability is also known as the intersection of events.

A Venn diagram is a useful visual tool for understanding intersections because it shows the overlap between sets or events.

There are several ways to find joint probabilities. The following sections discuss three standard methods: tables, independent, and dependent events.

Related post: Probability Definition and Fundamentals

Joint Probability Table

When dealing with multiple events, creating a table to organize the likelihoods can be helpful. A joint probability table lists the chances of event combinations at each row and column intersection.

Remember how ∩ represents an intersection? That makes sense in a table!

For example, suppose a survey asks people about their favorite color and animal. The researchers organize the results in the table below:

	Cat	Dog	Other
Red	0.10	0.05	0.03
Green	0.08	0.12	0.05
Blue	0.03	0.06	0.02

We want to find the likelihood that someone chooses red as their favorite color and a dog as their favorite animal. We can locate the intersection of the “Red” row and the “Dog” column, which is 0.05.

Therefore, the joint probability is:

P(Red ∩ Dog) = 0.05

If you have a contingency table that displays frequencies rather than likelihoods, you can use it to calculate joint probabilities. For instance, the previous table might have started as a regular contingency table. Learn how in my post, Using Contingency Tables to Calculate Probabilities.

Now let’s see how to calculate joint probabilities when you know the event likelihoods.

Joint Probability Formula for Independent Events

When two events are independent, the occurrence of one event does not affect the chances of the other event occurring. In this case, we can find the joint probability by multiplying the likelihood of one event by the likelihood of another.

The joint probability formula for independent events is the following:

P(A ∩ B) = P (A) * P(B)

For example, suppose we have a coin that we flip twice. We want to find the chances of getting heads on both the first and second flips. Because each flip is independent, the probability of the first heads is 1/2, and the likelihood of heads on the second flip is also 1/2. Therefore, the joint probability is the following:

P(H1 ∩ H2) = P(heads on first flip) x P(tails on second flip)

= 1/2 x 1/2

= 1/4

Similarly, the joint probability of rolling two sixes on six-sided dice is the following:

P(6₁ ∩ 6₂) = P(6 on first roll) x P(6 on second roll)

= 1/6 X 1/6 = 1/36

Related post: Independent Events

Formula for Dependent Events

We can use the general multiplication rule to calculate joint probabilities for dependent events. This rule allows us to factor in how the occurrence of one event affects the likelihood of the other event. Learn more about the Multiplication Rule.

The joint probability formula for dependent events is the following:

P(A ∩ B) = P(A) * P(B|A)

Here, P(A) represents the chances of event A occurring, while P(B|A) represents the conditional probability of event B occurring, given that event A has already happened. By multiplying these two likelihoods, we can calculate the joint probability of both events coinciding.

To solve this type of problem, you must know how the first event affects the likelihood of the second event.

Example

Suppose we need to calculate the likelihood of drawing two aces consecutively from a standard deck of 52 cards when we don’t replace the cards. Initially, the deck contains four aces, so the likelihood of drawing an ace on the first draw is 4/52 or 1/13. If we draw an ace (event A1), only three aces and 51 cards remain in the deck. Consequently, the conditional probability of drawing another ace (event A2) is now 3/51.

Using the general multiplication rule, we can find the joint probability of drawing two aces in a row:

P(A1 ∩ A2) = P(A1) * P(A2|A1)

P(A1) = 4/52 = 1/13

P(A2|A1) = 3/51

P(A1 ∩ A2) = (1/13) * (3/51) = 3/663 = 1/221

So, the joint probability of drawing two aces in a row is 1/221 or 0.0045.

In conclusion, joint probability is a powerful tool in statistics. They can model complex systems and help us make more informed decisions. Choosing the correct method to calculate them depends on the specific problem at hand.