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"Either/Or" Probability for Non-Overlapping Events

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Hi. This tutorial covers "either/or" probability for non-overlapping events. All right. So let's start with a chance experiment.

So while sitting on my balcony, I observed a car at an intersection, either going straight, turning right, or turning left. So this is a Google satellite picture of the intersection I live on. That's my building there.

So what I do is-- the balcony faces this way. So I watch a car travel down Grand Avenue here. And as it approaches the intersection, it can either go straight. It can turn right. Or it can turn left. So I'll that event L be that the car turned left and let event R be the car turned right.

So let's consider the outcomes of this experiment by using a Venn diagram to make the sample space. So with the Venn diagram I'm going to start with my box. And I'm going to let a circle represent event L and a circle represent event R.

OK. So if we start with event L, this would be the event-- this would represent anybody that makes a left-hand turn. Now, I also want to use a circle to represent event R. So the issue now is, should I make the circles overlapping, or should I keep them separate? So are they overlapping events or non-overlapping events?

So the way that I observed, can a car both turn right and left at the same time? And the answer to that is no. So what I'm going to do is I'm going to draw my circle representing the right-hand turns over here. So that's going to be right.

So are these of events overlapping? The answer there is no. And then everybody outside of the circles, two circles, would be the people that turned straight-- or excuse me-- that go straight.

So after performing the chance experiment many times, I found that 18% of cars turned left, and 7% of cars turned right. So if I use these to represent my probabilities, I would know that the probability of L would be 0.18, 18%. And then the probability of R would be 0.07.

OK. So now the question is, what is the probability of a car turning right or left? So if I denoted that probability, I would say the probability of L or R. I could also just write it out in words. So I could say L or R. Either of those are equivalent.

So what I'm going to propose-- it would just make sense just to add these two up. If 18% of people went left, 7% of people went right, then if you add those, that would be 25% went either left or right. Since there aren't any outcomes in common, it would seem logical that we could just add those two up. So we'd say that the probability of L plus the probability of R would be 0.18 plus 0.07, which would equal 0.25.

So what we just applied there is what we call the Special Addition Rule. The Special Addition Rule says that if events A and B are non-overlapping-- otherwise known as mutually exclusive-- then the probability of A or B is simply the probability of A plus the probability of B. So, again, special means that we can only use it under specific circumstances. So the condition for you to be able to use the Special Addition Rule is that A and B have to be non-overlapping.

OK. So if we look at an example of this, draw a card from a deck is the chance experiment. Let event A be the card is a club and event B be the card is a spade. So if we want-- and then we're trying to figure out the probability of A or B, so getting either a club or a spade.

OK. So to use the Special Addition Rule, we have to first decide whether or not the events are non-overlapping. So can you get a card that is both a club and a spade? No, you can't. So these events are non-overlapping, or mutually exclusive.

So we can just say that the probability of A or B equals the probability of A plus the probability of B. So the probability of A, the probability of getting a club, is 13 out of 52. The probability of getting a spade is the same probability, 13 out of 52. So if we add those two up, we end up with 26 out of 52, which is simply 0.5. So there's a 0.5 probability of drawing a card and getting either a club or a spade.

So this is a good application, again, of the Special Addition Rule. The events were non-overlapping. So we could simply just add the two probabilities.

So that has been the tutorial on "either/or" probability for non-overlapping events. Thanks for watching.