# Complementary Outcomes

Reference > Mathematics > ProbabilityTwo outcomes are complementary if they are the only possible outcomes for an event. For example, in flipping a coin, Heads and Tails are complementary, because they are the only possible outcomes (technically, it's possible for a coin to land on its edge, but we generally treat a coin toss as having two possible outcomes).

Another example would be, if you roll a six-sided die, getting an even number and getting an odd number are two complementary outcomes, because every possible result is either an even number or an odd number.

If you add together two complementary outcomes, what will the result be?

Remember what we said earlier in this unit about guaranteed outcomes? They have a probability of one. Thus, if there are only two possible outcomes, adding them together will give you one.

Look at the die-rolling example above. What is the probability of rolling an even number?**Example One**

If I pull a number from a hat, the probability that the number will be a prime is

*not*be prime?

**Solution**

Being prime and

*not*being prime are two complementary outcomes; either a number is prime or it isn't. Thus, if P(N) is the probability of not being prime, then

This example gives us a very useful formula: if A and B are complementary outcomes, then

P(A) = 1 - P(B)

It also helps us see that if we need to find the probability of something NOT happening, it can be easier to find the probability of it happening, and then subtract that probability from one.

**Example Two**

Ignoring leap year, what is the probability that someone's birthday is not in December?

**Solution**

Instead of adding the probabilities of the birthday being in January, February, etc. We remember that having a December birthday and NOT having a December birthday are complementary outcomes. Thus, if we find the probability of being born in December, we can subtract that from one to get the probability of NOT being born in December.

**Example Three**

I have a loaded six-sided die (loaded means that it is not a "fair" die; the probabilities are not equal). The probability of rolling a one is

**Solution**

Since 1, prime, and composite cover all the possibilities (one is neither prime nor composite), the sum of all these probabilities have to equal 1.

**Example Four**

I am three times as likely to pick an odd number out of the hat as I am to pick an even number. What is the probability I will pick an odd number?

**Solution**

We'll assume that all the numbers in the hat are integers, and therefore either odd or even. P(O) = 3P(E), and P(O) + P(E) = 1.