This video is provided as supplementary
material for courses taught at Howard Community College and this video is going to be about the
probability of mutually exclusive and non-mutually exclusive events, and this topic is a the whole lot easier
than it sounds. So let’s get started. I’ve got a die. It’s a cube, so it has six sides. One side has a ‘1’ on it, one dot, then there’s two dots, three dots, four, five and six. And I’ve got two events. Event A is throwing either a ‘1’ or a ‘3’ and event B is throwing either a ‘2’ or
a ‘4’. Now I can figure out the probability of each of these events. The probability of event A is the number of ways that it could
happen, and since there are two different ways it could happen, either a ‘1’ or a ‘3’, that will be 2 divided by the number of different outcomes, which
is 6. So the probability of A is 2 over 6, or 1/3. I can do the same thing for the
probability of B. It’s also got two different ways of happening — either a ‘2’ or a ‘4’ — so the probability of B is also going to be 2 over 6. or 1/3. If I want to find the probability of either A or B happening, then I can think of that as the
probability of A union B. Now let’s just work this out logically,
and then come up with a formula for it. So the probability of A union B means I’ve got to get either a ‘1’ or a ‘3’ or a ‘2’ or a ‘4’. Now there are 4 different ways that this
could happen – ‘1’, ‘3’, ‘2’, and ‘4’ — so the probability would be 4 over 6, the total number of outcomes. I could reduce that to 2/3. And realize that’s just the same as the probability of A plus the probability of B. So if I want a general formula, I can say the probability the union of A and B, the probability of A happening
and the probability of B happening is going to equal the probability of A plus the probability of B. Now if this works because A and B are what we call mutually
exclusive events. I’ll write that… mutually exclusive events. Mutually exclusive events are events that don’t have anything in
common. I can think of this is a Venn diagram.
If I have Venn diagram, I would have two circles which don’t
overlap. One would be the circle for event A, with a ‘1’ and a ‘3’ in it. The other would be the circle for
event B, with a ‘2’ and a ‘4’. So if we have mutually exclusive events, we can just add up to their individual
probabilities and get the probability of their union. Now let’s compare that to a different situation. So now I’ve got event A, and event A
is a ‘2’, a ‘4’ or ‘6’, and event B would be throwing a ‘4’, a ‘5’,
or a ‘6’. Now I’ve got some overlap here. Both of these events have a ‘4’ in them
and a ‘6’ in them. In terms of a Venn diagram, — I’ll draw a Venn diagram down here — I would have two overlapping circles. Event A is going to have a ‘2’, and then the overlapping area
will have a ‘4’ and a ‘6’, because event B is ‘4’ ‘5’ and ‘6’. So I’ve got a ‘4’ and a ‘6’ in both of those area. Now let’s look at their individual
probabilities. and then try finding the probability of A or B. So the probability of A is going to be the number of ways A
could happen. There are three different ways A could happen. So that’s three over six, or 1/2. The probability of B is going to be the number of ways
B could happen. There are three different ways
B could happen, so that’s also three over six or 1/2. Now if I use the formula for
mutually exclusive events, I would say that the probability of A union B — A or B — equals the probability of B plus the probability of B. Now if I applied that formula, what that
means is I would be adding the probability of A, which is 1/2, and the probability of B, which is 1/2… I’ve be adding 1/2 and 1/2 and that equals one, or 100 percent, and that’s like saying that there’s a 100 percent certainty that either A or B will happen. But there isn’t, because I might throw a ‘1’, which is not in event A or event B. Or I might throw a ‘3’, which is not in event A or event B. So there can’t be a 100 percent
probability. Here’s what the problem is. Because I have that overlap, I’m actually including the probability of a ‘4’ or a ‘6’ two times, one with event A and one with event B. So what I want to do is get rid of one of
those times when I have the probability of ‘4’ or ‘6’. So what I’m going to do is take this formula,
the probability of A union B equals the probability of A plus the
probability of B, and I’m going to subtract the probability of that intersection the ‘4’ and the ‘6’.
So I’m subtracting the probability of A intersect B. Let’s work that out with the numbers we have. So the probability of A was 1/2.
I’m gonna use 3/6, that will make it easier to do this. And the probability of B is also 3/6. Let’s find the probability of A intersect B.
Well, A intersect B is ‘4’ and ‘6’. So that’s two different ways that A intersect B could happen. So that means I’m subtracting 2 divided by 6. When I do that, I’m going to get 3/6 plus 3/6 minus 2/6. That works out to 3 plus 3 which is 6, minus 2, which is 4 over 6. And that reduces down to 2/3. Let’s look at this a different way and make
sure it works. If I want the probability of either A
or B, that means I’m looking for the
probability of a ‘2’ or a ‘4’ or a ‘5’ or a ‘6’. Those are really the numbers I’m dealing with.
There are 4 ways that could happen. There are 6 different total outcomes. So the probability of A or B of ‘2’, ‘4’, ‘5’ or ‘6’ happening is just 4 over 6, or 2/3, which is what I get with this formula. So here’s what that means in terms of
how much you might have to memorize. If I have mutually exclusive events, there is no
overlap. Since there is no overlap, the
intersection of those two events is zero. So I can use this formula. I can say that the probability of those
mutually exclusive events of A or B, is the probability of A plus the probability of B minus the probability of their
intersection, which is zero. iIf I have non-mutually exclusive events, like the ones i have here, with a ‘2’, ‘4’ and ‘6’ and then ‘4’, ‘5’ and ‘6’, once again I’ll use this formula — the
probability of A or B is equal to the probability of A plus
the probability of B minus the probability of the intersection of A and B. So that’s basically how this works. Run through this a couple of times.
It seems like it’s confusing until you just work through it and
realize how logically it works out with actual
numbers. So give it a try. Take care. I’ll see you next time.