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Complement of an Event

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In this tutorial, you're going to learn about what a complement of an event is. Specifically you will focus on:

- Complement of an Event

Complement is not the same way as saying something nice. This is complement with an e. Complement in a mathematical sense is actually just the event not happening.

Look at the roulette wheel and the probability that you'll land on a green sector when you throw the ball in there.

There are two green sectors out of 38, all of which are equally likely. The probability is 2/38.

What about the probability of getting something that isn't green? Count everything that's not green and you end up with 36 outcomes. 36 out of 38. It seems like there's a relationship between those two numbers.

An event not occurring is called a **complement of an event**. 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.

**Complement of an Event**

All outcomes not in the given event.

There are lots of different ways to notate this.

If you write this A with an apostrophe, it’s called A prime. 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. There's not really a whole lot of consistency among different textbooks or among the mathematical community..

Look back to the roulette wheel, the probability of landing on green was two out of 38.

The probability of the complement of green, which was landing on anything besides green, was 36 out of 38. There seems to be a relationship here. If you add 2/38 to 36/38 you get 1. Was there a way to calculate 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. Probability of green complement is equal to 1 minus the probability of green. You could have just said 1 minus 2/38, and I could have obtained my 36/38 that way.

To find the probability of the complement of each of these events, find the probability of the event first, and then subtract from one. This is 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? 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 of not flipping tails is 1/2.

Flip a coin 10 times. What's the probability that at least one of these flips is a head? Think about it using complements. What would be the complement of getting at least one head? The complement would be no heads.

The probability of no heads is so much easier to calculate than the other one. It's 1 out of 1,024. You don't need to know how to calculate that. 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.

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. 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.

Source: This work adapted from Sophia Author Jonathan Osters.