Probability rules

1. Complement rule:

Let A be an event and its complement. Then the complement rule is:

In words, the probability of the occurrence of any event equals one minus the probability of the occurrence of its complementary event.

Example:

A club has a membership of six men and four women. A three-person committee is chosen at random. What is the probability that at least one woman will be selected?

Solution:

Let A - be the event “at least one woman will be selected”. We will start solution by computing the probability of the complement: -“no woman is selected”, and then using the complement rule will compute the probability

of A.

And therefore the required probability is .

2. The addition rule of probabilities:

Let A and B be two events. The probability of their union is

Example:

The probability that a randomly selected student from a university is a senior is 0.18, a business major is 0.14, and a senior and a business major is 0.04. Find the probability that a student selected at random from this university is a senior or a business major.

Solution:

Let A - be the event “Chosen student is a senior student” and B the event “Chosen student is a business major student”. Thus we have

, and

The required probability is .

Example:

Suppose that in a community of 400 adults, 300 bike or swim or do both,

160 swim and 120 swim and bike. What is the probability that an adult selected at random from this community bikes?

Solution:

Let A be the event that person swims and B be the event that he or she bikes, then . Hence the relation implies that .

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