CK-12 Probability and Statistics - Advanced

http://www.ck12.org Chapter 3. An Introduction to Probability

The Complementary Rule

The sum of the probabilities of an event and its complement must equal 1.

P(A)+P(A′) = 1

As you will see in the following examples below, it is sometimes easier to calculate the probability of the complement
of an event rather than the event itself. Then the probability of the event,P(A), is calculated using the relationship:

P(A) = 1 −P(A′)

Example:

If you know that the probability of getting the flu this winter is 0.43, what is the probability that you willnotget the
flu?

Solution:

First, ask the question, what is the probability of the simple event? It is

P(A) ={you will get the flu}= 0. 43

The complement is

P(A′) ={you will not get the flu}= 1 −P(A) = 1 − 0. 43 = 0. 57

Example:

Two coins are tossed simultaneously. Here is an event:

A:{observing at least one head}

What is the complement ofAand how would you calculate the probability ofAby using the complementary
relationship?

Solution:

Since the eventAis observing all simple eventsA={HH,HT,T H}, then the complement ofAis defined as the
event that occurs whenAdoes not occur, namely, all the events that do not have heads, namely,

A′={observe no heads}={T T}

We can draw a simple Venn diagram that showsAandA′in the toss of two coins.