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
James Walsh

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

Expert Contributor
Will Welch

Will has a doctorate in chemistry from the University of Wyoming and has experience in a broad selection of chemical disciplines and college-level teaching.

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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Additional Activities

Identifying Independent vs. Dependent Events

In order to use the joint probability formula, you need to have two (or more) independent events. This means that the outcome of one event does not effect the probability distribution of another. If the events do effect one another, they are dependent.

Decide if the following events are independent or dependent:

  1. You spin a roulette wheel twice.
  2. You work with nine other people and you draw names for a gift exchange. You go first and your coworker goes second.
  3. You win at bingo twice in a row.
  4. You are dealt a pair of twos from one full deck of cards in blackjack. You ask for another card. It is a two.

Answers and explanations:

  1. The number the ball falls on the first time does not effect the outcome of a second spin, so the two events are independent.
  2. When you draw first, your odds of drawing any one name is 1 in 10. When your coworker draws, they cannot draw who you drew--the odds of that are zero. Also, their odds of drawing anyone left are one 1 in 9. So the outcome of the first drawing does effect the probability distribution of the next drawing.
  3. After each bingo game, all of the numbers are placed back into to the tumbler, so every game starts the same. Your odds of winning a second time are not effected by the fact that you won the first time.
  4. There are only 4 twos in a deck. If neither of your two initial cards had been a two, your chances of drawing a two afterwards would have been 4 in 50 or 1 in 12.5. Since you did get two twos, there were only two left over out of the 50 remaining cards, so the odds of getting the second two were 2 in 50 or 1 in 25. The first event did affect the probability of the second event.

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