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Complement of an Event

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Source: M&M's; Public Domain: http://en.wikipedia.org/wiki/File:Plain-M%26Ms-Pile.jpg

This tutorial discusses the complements of events. Now, a complement of an event is the set of all the outcomes that are not in the event, kind of the opposite, the ante event.

So here, rolling an even number-- the complement is going to be whatever is not rolling an even number. And everything that's not even is odd. So the complement of this event of rolling even number is going to be rolling an odd number, because all the even numbers-- 2, 4, 6-- everything that's not in that-- 1, 3, 5-- those are classified as odd.

Now, in our second example-- getting dealt a diamond. So for our complement, it's getting dealt anything that's not a diamond. So our complement is going to be getting dealt a spade, a club, or a heart. Anything that's not a diamond is the complement.

And the last one-- getting a primary color M&M, so red, blue, and yellow. We have a nice little snapshot of M&M's in case you can't remember M&M colors-- so everything that's not red, not yellow not blue. So for that, it would be getting a brown, getting a green, or getting an orange.

So that's our summary for our complements and our events. Now, here, we're going to talk a little about the probability of complements.

Now, the probability of a complement is 1 minus the probability of the event. There's a lot of different ways of representing this. You might see it any of these ways.

We could say the probability of the complement by using A prime. So that little prime there is telling us it's the complement. And that's the same as 1 minus the probability of the event.

So the key thing about all of these forms here is when we're talking about the probability of event, we're going to write that with P of A-- so probability of the event, whatever it is. So you could fill that in, and we're going to do that in a minute.

What changes is the way you write the word complement in math kind of notation. So here, you could refer to a complement as A prime, as A with a bar on top, or as A with a C up in the air as a superscript. I like to use this form the most often. I see this one a lot. I don't see this one as much, but it's good to be aware of it.

So here, our first event is rolling an even number-- so the probability of rolling an even number. And we have talked about that before-- the probability of rolling an even number. So we have six numbers possible. And 2, 4, and 6 are even, so that's three even numbers. So the probability of rolling an even number is 3/6, which is 1/2, which is 0.5.

The complement of that is going to be the opposite. It's rolling an odd number. Now, you can sometimes in simple examples calculate the complement out right away.

So our complement is rolling an odd number. And I know that's 1, 3, and 5. -- So I know that I should be finding 3/6, or 0.5. Let's test out our formula.

So the probability of my complement-- the probability of getting an odd-- is going to be equal to 1 minus the probability of getting an even. So the probability of the complement is going to be 1 minus the 0.5. And that is, in fact, 0.5. So our formula works. It's true.

Now, this was a pretty simple example. So we could do it both ways. Sometimes it's going to be a lot easier to find the probability of the original event and then work through it this way to find the probability of our complement.

You're going to notice that the probability of the original event and the complement are always going to add up to 1. Because just like before when we were talking about probability distributions, when you add up all of the possibilities total, those probabilities should all equal 1. Because the complement is everything that's not in the event, between those two pieces you should cover all of the possible outcomes. So the probability of both of those together should equal 1.

This has been your tutorial on complements.