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Complement of an Event
Common Core: S.CP.1

Complement of an Event

Author: Jonathan Osters

Calculate the probability of a complement in a given situation.

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In this tutorial, you're going to learn about what a complement of an event is. Complement is not the same way as saying something nice. This is complement with an e. And complement in a mathematical sense is actually just the event not happening.

So let's take a look at the roulette wheel. So probability that you'll land on a green sector when you throw the ball in here. Well, there are two green sectors out of 38, all of which are equally likely. So the probability is 2/38.

Now, what about the probability of getting something that isn't green? Well, we counted up two green ones, which means if we went through and counted everything that's not green, just counted it all up, 1, 2, 3, 4, 5, and so on, we would end up with 36 outcomes that weren't green. So 36 out of 38. Hm, seems like there's a relationship between those two numbers.

An event not occurring is called a complement of an event. So, for instance, the complement of the event landing in a green sector on the roulette wheel is landing in a sector that's not green. Now, just a quick note on notation, for some reason, there are lots of different ways to notate this. If you write this A with an apostrophe, we call it A prime. Sometimes that's called the complement of an event.

You can also use a bar. You can also use a c that sort of looks like an exponent. You can also use a tilde. For some reason, there's not really a whole lot of consistency among different textbooks, or among the mathematical community on this one.

You can also use this particular symbol. It sort of looks like a sideways l. Or the ever popular, just writing the word not in front of it. I'm going to use one with the c.

So taking a look back to the roulette wheel, the probability of landing on green was two out of 38. And the probability of the complement of green, which was landing on anything besides green, was 36 out of 38. I really do feel like there's a relationship here. In fact, if you add 2/38 to 36/38 you get 1. In fact, was there a way I could have calculated this 36 out of 38 without counting up the 36 non-green sectors?

The probability of the complement of an event is 1 minus the probability of the event. So probability of green complement is equal to 1 minus the probability of green. I could have just said 1 minus 2/38, and I could have obtained my 36/38 that way.

So find the probability of the complement of each of these events. Pause the video and scribble this down off to the side. It might be helpful to find the probability of the event first, and then subtract from one. So what you should have come up with is this, the probability of rolling a six on a die is 1/6. Therefore, the probability of not rolling a six is 5/6.

How about spinning red on a roulette wheel? Well, there are 38 sectors, 18 of which are red. 1 minus 18/38 is 20/38. Flipping tails on a fair coin, the probability of flipping tails is 1/2. The probability, therefore, of not flipping tails is 1/2.

Suppose you flip a coin 10 times. What's the probability that at least one of these flips is a head? Wow, this is a hard probability to calculate, because you might get one head, and it could come in really any order. It might be the first one or the last one. Or you might get two heads, and those can come in any order.

This is a hard probability to calculate directly, but think about it using complements. What would be the complement of getting at least one head? Well, the complement would be no heads.

Now, the thing about this is the probability of no heads is so much easier to calculate then the other one. In fact, it's 1 out of 1,024. You don't need to know how to calculate that. I'll just tell you that that's what it is. Therefore, the probability of at least one heads coming up on 10 flips is 1 minus the probability of no heads, so 1,023 out of 1,024.

And this leads us to an important finding within the context of real life problems. Whenever you're asking for the probability of at least one, something happens at least once, it's 1 minus the probability that it doesn't happen at all. At least once and not at all are complementary events. So probability of at least one is 1 minus the probability of none.

And so to recap, the complement of an event consists of all the outcomes that aren't in that particular event. And the probability of a complement of an event is 1 minus the probability of the original event. Also, you might want to keep in mind, because sometimes it's easier to calculate the complements probability and then subtract from one, like in the coin flipping example, than it is to calculate the probability that's being asked for directly. Good luck, and we'll see you next time.

Terms to Know
Complement of an Event

All outcomes not in the given event.

Formulas to Know
Complement of an Event

P left parenthesis A to the power of c right parenthesis space equals space 1 space minus space P left parenthesis A right parenthesis

Probability of "At Least One"

P left parenthesis " a t space l e a s t space o n e " right parenthesis equals 1 minus P left parenthesis " n o n e " right parenthesis