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"Either/Or" Probability for Overlapping Events
Common Core: S.CP.7

"Either/Or" Probability for Overlapping Events

Author: Katherine Williams
Description:

Calculate an "either/or" probability for overlapping events.

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Tutorial

Source: Graphs and tables created by Katherine Williams

Video Transcription

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This tutorial covers the general additional rule for probability. It's helping us to find out what's the probability of either event A or event B happening. So it is finding the either or probabilities.

Now the great thing about the general rule is that it works even if A and B have overlap. So the specific addition rule only works if there's no overlap. But this one allows for that.

Now our formula for the probability of A or B happening is the probability of A plus the probability of B, minus the probability of A and B. So that last part here, that is telling us what's overlapping. So when we add the probability of the two events happening, and then we subtract out the overlap, we were able to get the events of either one or the other.

Let's look at an example. Here, we have a problem involving people who are liking hamburgers and fries. And we have a Venn diagram in the corner. And we want to know what's the probability of liking hamburgers or fries.

So first, we need to find the probability of event A. So the probability of liking hamburgers. There are 220 people out of the 395 that like hamburgers. So that's the probability of A.

Now the probability of B is the probability of liking fries. So total there are 210 out of the 395.

Now the final part is to subtract the overlap. And these two events do overlap. There are 50 people who like hamburgers and fries. So we're going to subtract the 50 people who like hamburgers and fries.

And then when we compute all of this together, 220 plus 210 gives us 430, then we've accidentally double counted these people in the middle. So we're going to subtract 50. So we're not double counting anymore, we're subtracting the overlap, and we get 380.

And we can see by looking at our information that there are 395 people total. And of those 395 people, there are everybody in these circles, except for these 15 out here who don't like either. So there's 380 out of the 395 total that like hamburgers or fries.

We'll look at another example of a two-way table. So here in this contingency table, it's showing us the likelihood of having a high, low, or medium salary, based on how much they paid for their MBA tuition. Now, what's the probability of having a low tuition or a medium tuition?

So first, we need to find the probability of a low tuition. So the probability of a low tuition is this part here.

Now B, the medium tuition, the probability of being a medium tuition, are all these three pieces, which adds up to 0.28. Now, we have to subtract out any overlap. And there are, in fact, no people who paid low and medium tuition. Doesn't make sense, they only went to the college once, so they can only be in one of those two groups.

So here, we're going to do 0.24 plus 0.28, and then minus nothing, because there's no overlap. I like to put minus zero there because it reminds me that I checked for it and that I didn't find any. But you don't have to write that at all. Then when I add these two probabilities together, I get 0.5. So you had a 52% chance of paying either a low tuition or a medium tuition.

Now, let's do another example. Here, we are finding the probability of paying a low tuition or having a low salary. Now, the low tuition, so we need the P of A.

So the probability of having a low tuition is this group here, which totalled 0.24, plus the probability of B. So plus the probability of having a low salary is another 0.24.

Now I need to subtract the probability of A and B. So is there anybody who had a low tuition and a low salary? And that is true-- there are people who did both, so minus 0.07.

So this time, we did have overlap. So there was something that we needed to subtract. When we add all these pieces together and subtract the overlap, we get 0.41. So there's a 41% chance of having a low tuition or a low salary.

This has been your tutorial on the general addition rule for probability.

Terms to Know
Either/Or Probability for Overlapping Events

The probability that either of two events occurs is equal to the sum of the probabilities of the two events, minus the joint probability of the two events happening together.  Also known as the "General Addition Rule".

Formulas to Know
Either/Or Probability for Overlapping Events

P left parenthesis A space o r space B right parenthesis space equals space P left parenthesis A right parenthesis space plus thin space P left parenthesis B right parenthesis thin space minus space P left parenthesis A space a n d space B right parenthesis