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

"Either/Or" Probability for Overlapping Events

Author: Ryan Backman
Description:

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

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Tutorial

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Hi. This tutorial covers either/or probability for overlapping events. Let's start with a situation here. So suppose a freshman from a local college is selected at random. Let event A be the event that a freshman takes Intro to Statistics. Let event B equal the freshman takes Intro to Psychology.

And then we'll let probability of A equal 0.18, probability of B equal 0.21. So can we determine A or B? OK, so remember that the special addition rule says that the probability of A or B equals the probability of A plus the probability of B. But now, remember, that only works if the two events are non-overlapping.

So let's consider these two events, event A and B. So can there be common outcomes to both event A and event B? Can a freshman take both Intro to Statistics and Intro to Psychology? And generally, yes. There might be outcomes that are common to both events. So this 18% might include people that are enrolled in both courses. This 21% might be-- might include people from both courses.

So if we were just to add these two probabilities up, we're going to end up double counting the people that are enrolled in both courses. So at this point, no, we cannot determine A or B. All right, so we need a new rule. So the special addition rule-- that's the one we just went over. If events A and B are non-overlapping, then the probability of A or B equals the probability of A plus the probability of B.

Now, the general addition real, notice, does not have a condition. This can be used in all cases. So the probability of A or B equals the probability of A plus the probability of B minus the probability of A and B. So if we think about A and B, that's going to include the probability of-- this is going to give you the probability of a student taking both classes, both statistics and psychology.

So if those students are included in both A and B, we need to subtract off A and B so that those people are not double counted. Now, also remember that, if events are non-overlapping, the probability of A and B is equal to zero. So if this equals zero, the general addition rule becomes the special addition rule, because if they're non-overlapping, A and B is equal to zero.

So if we think back to the example again. So why couldn't we calculate A or B? Because we didn't know what's the probability of a student taking both classes. So what we're going to do now is suppose that A and B equals point 0.08. So there's an 8% chance that students will take both of the two classes.

So I think it's helpful to visualize these probabilities in a Venn diagram, so let's draw a Venn diagram here. So what we're going to need is a circle for Intro to Statistics-- which we'll call that A-- a circle for Intro to Psychology. Since there is an overlap, I'm going to draw the circles overlapping.

So this is going to be psychology. We're going to put the 0.08 in the overlapping region. So that's 0.08. Now, A, this region here, are the students that take Intro to Statistics, but do not take Intro to Psychology. So remember that, altogether, 18% take statistics. So that means 10%-- only 10% take statistics, but not psychology.

Then, for only psychology and not statistics, that's going to be 0.13, or 13%, because these two then will now add up to that 21%. Now, if we want to figure out what's the probability that a student takes neither psychology nor statistics, all we need to do is add up the three numbers that are already in our Venn diagram-- so that's 31%-- and subtract off-- subtract that from 1 to get 0.69-- so 69%.

Now, if we want the probability of A or B, there's really two ways to do it. We can just add up these three outcomes or we can use our new rule, the general the general addition rule. So we knew that these added up to 0.31, so let's confirm that using the rule. So we'll take the probability of statistics plus the probability of psychology minus the probability of both.

So we're going to have 0.18 plus 0.21 minus 0.08. And to do that, let's just do those in the calculator. And that does end up giving us 0.31. So there is a 31% chance that a randomly selected student will take either stats or psychology, or both. OK, so let's do one more example.

So I usually like 70% of the movies I see. My wife likes 80% of the movies she sees. We both like 50% of the movies we both see. So if we see it together, there'll be a 50% chance that both of us like the movie. So if we notate these probabilities, let's call this the probability of A, where event A is the event that I like the movie.

Let's let the 80% be event-- or the probability of B. So B is the probability that my wife will like it. And 50% will be the probability of A and B. So if my wife and I see a movie, what's the probability that my wife or I will like the movie? So here we're looking for now the probability of A or B.

And they are overlapping events, because there are movies that we both like, so we'll use the general addition rule, which says the probability of A plus the probability of B minus the probability of A and B will equal the probability of A or B. So if we do that, this is going to be 0.7 plus 0.8 minus 0.5. And use the calculator. That's going to give me 1.

So it actually worked out pretty nicely here. There is 100% chance that either my wife likes it, or I like it, or we both like it. So since that was 1, there's 100% chance of that occurring. So that is the tutorial on either/or probability for overlapping events. Thanks for watching.

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