4.3 Probability of a Complementary Event

Learning Outcomes

Calculate the probability of a complementary event

Recall operations on fractions

Adding and subtracting fractions with common denominators

$\dfrac{a}{c}\pm \dfrac{b}{c}=\dfrac{a\pm b}{c}$

In the two equations below, note that this relationship is described in both directions.

That is, it is also true that

$\dfrac{a\pm b}{c}=\dfrac{a}{c}\pm \dfrac{b}{c}$

The second equation furthermore includes the fact that

$\dfrac{a}{a}=1$

Complementary Events

Now let us examine the probability that an event does not happen. As in the previous section, consider the situation of rolling a six-sided die and first compute the probability of rolling a six: the answer is P(six) =1/6. Now consider the probability that we do not roll a six: there are 5 outcomes that are not a six, so the answer is P(not a six) = $\frac{5}{6}$ . Notice that

$P(\text{six})+P(\text{not a six})=\frac{1}{6}+\frac{5}{6}=\frac{6}{6}=1$

This is not a coincidence. Consider a generic situation with n possible outcomes and an event E that corresponds to m of these outcomes. Then the remaining n – m outcomes correspond to E not happening, thus

$P(\text{not}E)=\frac{n-m}{n}=\frac{n}{n}-\frac{m}{n}=1-\frac{m}{n}=1-P(E)$

Complement of an Event

The complement of an event is the event “E doesn’t happen”

The notation $\bar{E}$ is used for the complement of event E. Other commonly used notations for the complement of E are E’ or E^c.
We can compute the probability of the complement using $P\left({\bar{E}}\right)=1-P(E)$
Notice also that $P(E)=1-P\left({\bar{E}}\right)$

example

If you pull a random card from a deck of playing cards, what is the probability it is not a heart?

Show Solution

There are 13 hearts in the deck, so $P(\text{heart})=\frac{13}{52}=\frac{1}{4}$ .

The probability of not drawing a heart is the complement: $P(\text{not heart})=1-P(\text{heart})=1-\frac{1}{4}=\frac{3}{4}$

This situation is explained in the following video.

Try It

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Module 4: Probability and Statistics