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And Probability for Independent Events
Common Core: 7.SP.7a 7.SP.8a S.CP.2 S.CP.8

And Probability for Independent Events

Author: Jonathan Osters

This lesson will explain the rule for finding the probability of two or more independent events all happening.

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Introduction to Statistics

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-In this tutorial, you're going to learn about "And" probability. "And" means that you're going to do A and B. Events A and B both have to be the case. And they're going to be independent.

There's different "And" probability formulas for independent events versus dependent events. This tutorial is going to deal with independent events. So let's take a look.

Suppose a coin is flipped-- this coin-- and this spinner is spun and apparently A and D are a lot more common than B or C. Since the outcome of the coin flip doesn't have any influence on the probabilities for the spinner, these events are called independent. Spinning the spinner wouldn't have any effect on the probability of the coin coming up heads. And knowing if the coin came up heads doesn't affect the probabilities for the spinner.

So we can create a tree diagram to show all the outcomes. Now there are two outcomes for the quarter and there are four outcomes for the spinner. Which means there are eight total outcomes. But they're not all equally likely.

So first off, these are the eight heads followed by A, heads followed by B et cetera. And the heads and tails each have a one-half probability. That the A, B, C and D don't all have the same probability. If you look closely at the spinner, A is the biggest sector, followed by D, and then B and C. In fact, the truth is that they are in this ratio. 35% A, 25% B, 10% C, 30% D.

And since it doesn't matter whether you got heads the first time or tails the first time-- the probabilities will remain the same-- We can apply them down here too.

Now all I want to know, what's the probability that you get both heads and sector D. Well, let's take a look. Suppose that we did this experiment 1,000 times. You would expect about 500 heads and 500 tails, if everything went according to your expectation. If the second group went according to your expectation, 35% of those 500 would say A. 25% of those 500 would say B. 10% of those 500 would say C. And 30% of those 500 would say D. Ultimately, you would end up with 175 of the 1,000 that went heads A, 125 of the 1,000 that went heads B.

Ultimately what I'm looking for is heads D, heads and D. 150 of the 1,000 that we did. Or, when you simplify 150 out of 1,000 you get 0.15.

Now I wonder is there another way to obtain 0.15. Well, look at the path that we traveled to get here. We went from heads down to D. And we had those probabilities of 1/2 and 0.3. And the probability of H and D was 0.15. Hopefully, what you're seeing is that these two values multiplied give us 0.15.

So what we have is a multiplication for independent events. The "And" probability, the probability of events A and B both happening, is the product of their individual probabilities. In this case, it was 1/2 for heads and 0.3 for D. And we multiply them and we got 0.15. This is a big rule we'll put majesty around it. It's a big deal I'd like you to remember.

So to recap, there's a special multiplication rule for finding "And" probability, which are also calling joint probability for two independent events. The probability that they both occur is equal to the product of their individual probabilities. So we talked about the special multiplication rule for independent events. There will be a different multiplication rule if the events aren't independent. Good luck, and we'll see you next time.

  • Special Multiplication Rule

    A way to approximate probability based on trials of chance experiments that mimic the real-life trials.