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# Complementary Outcomes

Reference > Mathematics > Probability

Two 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?
3
6
. What is the probability of rolling an odd number?
3
6
. What is the probability of rolling either an even or an odd?
3
6
+
3
6
=
6
6
= 1.

Example One
If I pull a number from a hat, the probability that the number will be a prime is
2
15
. What is the probability that the number will 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
2
15
+ P(N) = 1, so P(N) = 1 -
2
15
=
13
15
.

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.

P(D) =
31
365

P(not D) = 1 - P(D) = 1 =
31
365
=
334
365

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
1
7
. The probability of rolling a prime number is
1
2
. What is the probability of rolling a composite number?

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.
1
7
+
1
2
+ P(C) = 1. Thus, P(C) = 1 -
1
7
-
1
2
=
5
14
.

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.

Thus, 3P(E) + P(E) = 1, or 4P(E) = 1. This leads to P(E) =
1
4
, and P(O) = 1 -
1
4
=
3
4
.

## Questions

1.
I selected a card from the deck. What is the probability that it is not a king?
2.
When I get to the intersection, I will either turn left or right. The probability that I will turn left is
2
7
. What is the probability that I will turn right?
3.
On a loaded four-sided die, the probability of rolling a one is
1
3
and the probability of rolling an even number is
1
2
. What is the probability of rolling a three?
4.
What is the probability (ignoring leap years) of not being born on the first day of the month?
5.
I have 14 blue socks, 20 red socks, 18 yellow socks, and 30 green socks. If I randomly pick one sock, what is the probability that it is not green?
6.
I selected a card from the deck. What is the probability that it is not a spade?
7.
My unfair coin is twice as likely to be heads as tails. What is the probability that it will be tails?
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