Probability of complementary events

In this lesson, you will learn to compute the probability of complementary events. What are complementary events though?

Let A be an event. The complement of A consists of all outcomes in which event A does not occur.

The symbol that we use to denote the complement of A is A

A and  the complement of A are called complementary events

Sometimes, we need to compute the probability that an event A occur and the probability that event A does not occur.

For example, company x conducted a survey to find out whether or not cars built in the 1970 had defects in the engine.

The result revealed that 4 out of every 100 cars had a defect in the engine. What are the probabilities of the two complementary events?

The two complementary events for this experiment are the following:

Let A = the car selected has a defect in the engine.

Let A = the car selected does not have a defect in the engine.

Here are the probabilities

P(A) =
4 / 100
= 0.04

P(A) =
96 / 100
= 0.96

Notice that 0.04 + 0.96  = 1

In general the probability of complementary events will add up to 1.

P(A) + P(A) = 1

Notice also that event A and the complement of event A are mutually exclusive. This means that event A and the complement of event A have nothing in common.

This makes sense since it is not possible for a car to have a defect and not have a defect at the same time.

We can conclude that two complementary events are always mutually exclusive.

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