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Complement Rule for Probability

If P(A) is is the probability that event A happens then the probability that event A doesn't happen is 1-P(A).

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Identifying the Complement

Identifying the Complement 

The complement of an event is the sample space of all outcomes that are not the event in question. The complement of the event “a flipped coin lands on heads” is “a flipped coin lands on tails”. The complement of “A six-sided die lands on 1 or 2”. Is “A six-sided die lands on 3, 4, 5, or 6”. Complements are notated using the prime symbol ’ as in: \begin{align*}P(A^\prime)\end{align*} is the complement of \begin{align*}P(A)\end{align*}.

The probability of the complement of an event is always whatever probability it would take to reach 100%. If the probability of pulling a green marble out of a bag is 26%, then the probability of the compliment (pulling a not green marble) is 74%.

By convention, we most often see probabilities described as either a percentage or a fraction. Despite this, every calculated or experimental probability can be expressed as a value between 0 and 1 since percentages and fractions can all be converted to decimals and a probability must be between 0% and 100%. For example, a probability of \begin{align*}\frac{3}{4}\end{align*} or 75% could also be expressed as the decimal .75, and a probability of 20% or \begin{align*}\frac{1}{5}\end{align*} could be expressed as the decimal .20.

One benefit of viewing probabilities as decimals is that it is easy to calculate the complementary probability of a given event by subtracting the event probability (expressed as a decimal) from 1.

Finding the Complement 

What is the complement to the event “Brian chooses one of the 2 red shirts from his drawer containing 10 shirts”?

Credit: OpenClips
Source: http://pixabay.com/en/t-shirt-shirt-red-clothing-156081/?oq=red%20t
License: CC BY-NC 3.0

The complement would be the other possibility: “Brian chooses one of the not red shirts from his drawer.”

Calculating Probability 

1. If the probability of randomly choosing a Queen from a standard deck of 52 cards is .077, what is the probability of the complementary event?

The complement would be choosing a card that is not a Queen, and the complement probability would be the difference between .077 and 1:


Therefore, the if the probability of choosing a Queen is 7.7%, then the probability of choosing a card not a Queen is 92.3%

2. What is the probability of the compliment of the event: “Roll a standard die and get an even number”?

There are three even numbers on a standard die: 2, 4, and 6. That means that the probability that you do roll and get an even number is:

\begin{align*}P(even)=\frac{3}{6} \ or \ 50 \%\end{align*}

Therefore, the complement is:

\begin{align*}P(even^\prime)=1-.50=.5 \ or \ 50 \%\end{align*}

Earlier Problem Revisited

The complement of an event is the set of all outcomes that are not the event.


Example 1

If \begin{align*}P(X)=\frac{1}{6}\end{align*}, what is \begin{align*}P(X^\prime)\end{align*}?


Example 2

What is the probability of the complement to a probability of 74%?

The complement probability is \begin{align*}100 \% -74 \%=26 \%\end{align*}

Example 3

What is the complement to the event: “flipped coin lands on heads”?

"flipped coins land on tails"

Example 4

What is the probability of the complement of randomly choosing one of the 3 quarters from a set of 10 coins?

The event probability is \begin{align*}\frac{3 \ quarters}{10 \ coins}=\frac{3}{10}\end{align*}. The complement is \begin{align*}1-\frac{3}{10}=\frac{7}{10}\end{align*}.

Example 5

What is the percent probability of \begin{align*}Y^\prime\end{align*} if \begin{align*}\text{P(Y)}=\frac{1}{8}\end{align*}?

  1. \begin{align*}P(Y^\prime)=1-P(Y)=1-12.5 \%=87.5 \%\end{align*}


For problems 1 – 10, identify the percent probability of the complement of the described event.

  1. Roll a standard die once and get an even number.
  2. Pull a red card from a standard deck.
  3. Pull a face card from a standard deck.
  4. Roll two standard dice and get a sum greater than 9.
  5. Pull two cards from a deck, without replacement, get at least one face card.
  6. Roll a 10-sided die twice, get a 6 both times.
  7. The probability that a student in your class likes chocolate is 34%.
  8. Of the 76 students in your math class, 26 earned an A.
  9. 23% of million-mile cars are Toyotas.
  10. A candy machine has 24 green, 32 red, and 14 yellow candies in it. You choose a yellow candy.
  11. There are 150 students in your class, 40 have laptops, and 110 have tablets. 26 of those students have both a laptop and a tablet. What is the probability that a randomly chosen student has a tablet, given that she has a laptop?
  12. Roll two standard dice, and get 4’s on both, given that you know that you have already rolled a 4 on one of them.
  13. Draw two cards in a row, without replacement, that are the same suit from a standard deck.
  14. Roll of two standard dice once, getting a sum greater than 8, given that one of the dice is a 6.

Review (Answers)

To view the Review answers, open this PDF file and look for section 6.7. 

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complement A mutually exclusive pair of events are complements to each other. For example: If the desired outcome is heads on a flipped coin, the complement is tails.
Complement rule The Complement Rule states that the sum of the probabilities of an event and its complement must equal 1, or for the event A, P(A) + P(A') = 1.
enumerate Enumerate means to catalogue or list members independently.
Venn diagrams A diagram of overlapping circles that shows the relationship among members of different sets.

Image Attributions

  1. [1]^ Credit: OpenClips; Source: http://pixabay.com/en/t-shirt-shirt-red-clothing-156081/?oq=red%20t; License: CC BY-NC 3.0

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