Probability > Complementary events

## What are Complementary Events?

Complementary events happen when there are only two outcomes, like getting a job, or not getting a job. In other words, the complement of an event happening is the exact opposite: the probability of it **not** happening.

Venn diagrams are sometimes used to show complementary events. In the following Venn diagram, The complement of a set *A* is everything that is not in *A* [1]; it is represented by the blue region (the yellow region is set A:

## Examples of complementary events

- It rains or it does not rain.
- You pass your class or you don’t.
- Your dog bites the mailman, or it doesn’t.
- You win the lotto, or you don’t.
- You get married, or you don’t get married.

**The odds do not have to be equal. The important thing is that the two probabilities (the probability of an event happening and the probability of it not happening) must add up to 1. For example:**

- Rolling a die to see if you would get five is complementary; the only two outcomes are: getting a five (a 1/6 chance) or not getting a five (5/6 chance). 1/6 + 5/6 = 1.
- The odds of you winning the lotto might be one in a million; your odds of not winning then are 999,999 out of a million. The two probabilities equal 1: 1/1,000,000 + 999,999/1,000,000 = 1.

Complementary events must also be mutually exclusive events (events that can’t happen at the same time). This makes sense, as if you get a job, you can’t *not* get the job at the same time. You’ll sometimes see the complement written as [A’] pronounced “not A” or [A^{c}] pronounced “the complement of A’. They mean the same thing.

## The rule of complementary events

The rule of complementary events comes from the fact of the probability of something happening, plus the probability of it not happening, equals 100% (in decimal form, that’s 1). For example, if the odds of it raining is 40%, the odds of it *not* raining must equal 60%. And 40% + 60% = 100%.

**P(A ^{C}) + P(A) = 1**

You may also see this formula written like this:

**p(A) + p(A′) = 1**

which can be rearranged algebraically to read:

**p(A′) = 1 – p(A).**

**All three formulas are equivalent.** Which terminology (A′ or A^{c}) is used is up to the textbook author and teacher. For example, Business Statistics by Ken Black uses A’. Practitioner’s guide to Statistics uses yet another notation (̄A). Personally, I prefer A′, which I call “not A.” The probability of “not A”, I think, is easier to understand than “the complement” (either an event happens, or it does not happen).

Complementary in statistics can refer to the complement of an event A, which is the event that A does *not *occur. This is a more formal description of the above outline. However, it can also refer the complement of a function f(x), which is the function that takes on the value 1 when f(x) is 0, and vice versa. The connection between the two is is that they both refer to the opposite of an event.

Related article: Complementary sequences.

## References

[1] Venn Diagrams