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Probability of Event A OR Event B
Now that we’ve talked about the idea of mutually exclusive, we’re ready to talk about one of the major rules, the OR rule. This rule has two versions, a simple rule and a general rule. So the first thing I’ll say is that the simple rule, the essence of the simple rule is our shortcut approximation: in probability, the word OR simply means add.
Now to be a little more precise about this, if events A and B are mutually exclusive, then the probability of A or B equals simply the probability of A plus the probability of B. In other words the way to figure out the probability of OR Is simply to add the two individual probabilities.
This is the rule if the two events are mutually exclusive. A way to visualize this incidentally, two events are mutually exclusive what this means is that when we draw them as circles there’s gonna be no overlap. In other words there’s no place on this diagram where a point can simultaneously occupy circle A and circle B. That’s what it means to be mutually exclusive.
Well if we want to figure out the total area taken up by these two circles together, obviously that would just be the sum of the individual areas. And so, at a fundamental level, this is exactly why the word OR means to add. But what happens if they’re not mutually exclusive? Well if events A and B are not mutually exclusive, what this means is that there’s some overlap, some occasions when both A and B can happen together.
So let’s think about this now. This would be a diagram for two events that are not mutually exclusive. In other words, it is possible to land somewhere in this diagram in this blue region and be in a place where both A is happening and B is happening at the same time. So these are two events that are not mutual exclusive.
Well, let’s just think about this for a second. If we wanted the total area taken up by this whole thing, you want that area, well it’s not gonna work just to add up the area of circle A and the area of circle B. That’s gonna give us more area than we need, because there’s that overlap region, and really what would be happening if we just added probability of A, probability of B, that little overlap region in the middle, that would get counted twice.
And so that would be the thing that we’d have to correct for. So, because the overlap region gets counted twice, and we only want to count it once, we need to subtract that overlap region from the sum of the two individual regions. And therefore the rule is the probability of A or B is the probability of A plus the probability of B minus the probability of A and B.
This is a probability rule that is true 100% of the time. This is a very powerful rule to know. And if you can remember the diagram in the previous slide, that will help you remember why this particular rule has this particular form. So, overall there are two probability or rules. There’s the simple rule.
This only true for mutually exclusive events. The probability of A or B equals the probability of A plus the probability of B. Then there’s the general rule that is true 100 percent of the time. The probability of A or B equals the probability of A plus the probability of B minus the probability of A and B.
So I realize right now, this is very abstract. What we’re showing is all kinds of rules in this abstract, algebraic notation. In the next video, we’ll actually show an example of how this plays out in an individual problem.
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