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Overlapping Events

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Hi. This tutorial covers overlapping events. All right, so let's consider a chance experiment. A person is randomly chosen. His or her gender is noted. The person is asked if he or she is right or left-handed. So we're going to let event A be the event that a person is female and we're going to let event B be that-- be the event that the person is left-handed.

So let's take a look at the outcomes of this experiment by using a Venn diagram to make the sample space. So what I like to do is, again, start with the box here, the rectangle. Then what I'm going to look at is-- I have two events that can either be-- that can occur or not occur. So let's start with event A, the event that the person is female.

So I'll call this event A. Anybody within the circle represents somebody that's female. Outside of the circle would represent somebody that's not female. Now, the next circle I would need to draw would represent the people that are left-handed, and outside of that would be the people that are right-handed.

The question now is I want to make a circle for event B. Now, do I want it to overlap event A or do I want to be out here on its own not overlapping event A? So the question is, can a female also be left-handed? And of course, the answer to that is yes. So what I'm going to do is I'm going to draw my circle here to represent the set of all people that are left-handed.

So basically, this Venn diagram now displays four outcomes. The people outside of the circles would be right-handed males, the people inside of this region would be left-handed females, the people in just this area would be right-handed females, and the people in this area here would be left-handed males.

So a couple of definitions-- overlapping events are two or more events that share one or more outcomes. So because A and B overlapped here, because there were people that were both female and left-handed, A and B are considered overlapping events. Overlapping events can happen at the same time, so you can select somebody that is both female and left-handed at the same time.

And then what I wrote here kind of as an aside-- the probability of A and B must be greater than zero. So there needs to be a non-zero probability of both A and B occurring. Non-overlapping events-- sometimes another term for this is mutually exclusive events, so non-overlapping and mutually exclusive are synonymous.

And what those are two or more events that cannot share outcomes. So non-overlapping events cannot happen at the same time. So in this case, the probability of A and B is equal to zero. So non-overlapping events-- these two circles would have to be separated, each in their own area there.

So just to think about what that picture would look like, we'd have our sample space. We would have two events here. Those would be non-overlapping events. Non-overlapping events cannot be independent. If the events were independent, recall that the probability of A and B has to equal the probability of A times the probability of B.

So if we consider this to be event A and this to be event B, both of these have probabilities that are non-zero. But if we said that the probability of A and B has to equals 0, for something to equal 0 where two things are being multiplied, one of these have to be 0. If both of these have some probability of occurring, there's no way that either of these will equal 0, so this cannot equal 0. Because it cannot equal 0, non-overlapping events cannot be independent.

All right, so let's take a look at one last example here. So if we look at this chance experiment, a card is drawn from a deck. Event A, the card is a face card-- event B, the card is an ace. Are A and B overlapping or non-overlapping events? So basically, can we have an outcome that is in both A and B? So can we get a card that is both a face card and an ace?

In that case, no, you can't. Face cards are jacks, queens, and kings. A face card is not an ace, so there's no way you can get a card that's both a face card and an ace. So what we would say in this case-- A and B are non-overlapping events. So this has been your tutorial on overlapping events. Thanks for watching.