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Joint Probability: Definition, Formula & Examples

Joint Probability: Definition, Formula & Examples
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  • 0:00 What Is Joint Probability?
  • 1:17 Joint Probability Formula
  • 2:26 Another Example Using…
  • 3:21 Lesson Summary
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Lesson Transcript
Instructor: James Walsh

M.B.A. Veteran Business and Economics teacher at a number of community colleges and in the for profit sector.

Joint probability is the likelihood of two independent events happening at the same time. Joint probabilities can be calculated using a simple formula as long as the probability of each event is known. This lesson will illustrate the formula with examples.

What Is Joint Probability?

Joint probability is simply the likelihood that two events will happen at the same time. It's the probability that event X occurs at the same time as event Y. Sounds easy, right? Well, there are a couple conditions. One is that events X and Y must happen at the same time. Throwing two dice would be an example of that. The other is that events X and Y must be independent of each other. That means the outcome of event X does not influence the outcome of event Y. Our dice roll is again a good example of independent events, as the outcome of rolling one die has no influence on the outcome of rolling the other. If your first die comes up a 1, the probability of the second die is still a 1/6 chance for each number between one and six.

So what's an example of two events that are not independent? Well, how about event X is the probability there are clouds in the sky, and event Y is the probability that it rains. Even Wally the Wacky Weatherman (who is wrong a lot!) knows that rain comes from clouds. So rain can only fall when there are clouds in the sky. That means the presence of clouds will influence the chances of rain, and that means these two events are not independent!

Joint Probability Formula

Jill is playing a board game. It is her turn, and she wants to roll exactly a twelve to reach her goal. The only way to get that twelve is to roll a six on each die. Since we already know that rolling two dice are independent events, we can use the joint probability formula to calculate her chances for success. Here is the formula:


The joint probability formula
Joint Prob Formula


If the probability of rolling a six on one die is P(X) and the probability of rolling a six on the second die P(Y), we can use the formula P(X,Y) = P(X) * P(Y) . Since the dice have six sides, and the probability of any side coming up is equal, P(X) and P(Y) both equal 1/6. Thus, the formula looks like the one appearing on your screen right now and eventually results in a probability of 2.8%.


Use the formula to calculate the probability of two sixes when rolling dice. Not very good!
Joint Prob 3


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