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
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!
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Jill will have to close her eyes and wish really hard for those sixes!
Another Example Using the Formula
Ike likes to invest his savings in the stock market. Right now he owns two stocks, company X and company Y. He's just heard some optimistic news about the economy and thinks that stocks are headed up today. He learns that the probability of stock X going up is 80% and stock Y going up is 60%. He wonders what the probability is that both of his stocks go up. He can use the joint probability formula to find out! This scenario meets the conditions for the formula since it's the same day for both stocks, and one stock going up or down has no influence on what the other does. That makes them independent events. He converts the percentage probabilities to decimals and plugs them into the formula.
Use the formula to solve the problem
We see that it's 80% x 60%, which comes out to be a probability of 48%. Now those are almost 50/50 odds. Ike thinks this may be a lucky day!
Joint probability is the likelihood of two independent events happening at the same time. The two events must be independent, meaning the outcome of one event has no influence over the outcome of the other event. The joint probability formula is used by multiplying the probability of each independent event by the other. For events X and Y with probabilities P(X) and P(Y), the probability of both occurring is P(X, Y). Then:
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:
You spin a roulette wheel twice.
You work with nine other people and you draw names for a gift exchange. You go first and your coworker goes second.
You win at bingo twice in a row.
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:
The number the ball falls on the first time does not effect the outcome of a second spin, so the two events are independent.
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.
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.
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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