2.8. 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 .

Exercises

1. The probability that a randomly selected elementary school teacher from a city is a female is 0.68, holds a second job is 0.42, and is a female and holds a second job is 0.29. Find the probability that an elementary school teacher selected at random from this city is a female or holds a second job?

2. It was estimated that 35% of all students were seriously concerned about employment prospects, 20% were seriously concerned about grades, and 15% were seriously concerned about both. What is the probability that a randomly chosen student is seriously concerned about at least one of these two things?

3. It was found that 45% of students think that professors must be “more tolerant” to the students. If a student is selected randomly what is the probability that he or she will disagree or have no opinion on the issue.

4. In a statistics class there are 18 juniors and 10 a seniors; 6 of the seniors are females, and 12 of the juniors are males. If a student is selected at random, find the probability of the following:

a) A junior or a female

b) A senior or a female

c) A junior or a senior

5. If a die is rolled three times, find the probability of getting at least one 6.

6. If a die is rolled three times, find the probability of getting at least one even number.

7. A number is selected at random from the set of natural numbers

. What is the probability that it is divisible by 4 but neither by 5 nor by 7?

8. There are four tickets numbered 1, 2, 3, and 4. Suppose a two-digit number will be formed by first drawing one ticket at random and then drawing a second ticket at random from remaining three.(For instance, if the first ticket drawn shows 4 and the second shows 1, the number recorded is 41.) List the sample space and determine the following probabilities

a) What is the probability of getting an even number?

b) What is the probability of getting a number larger than 20?

c) What is the probability that obtained number is between 22 and 30?

9. A three-digit number is formed by arranging the digits 1, 5, and 6 in a random order.

a) List the sample space.

b) Find the probability of getting a number larger than 400.

c) What is the probability that an even number is obtained?

Answer

1. 0.81; 2. 0.40; 3. 0.55; 4. a) 6/7; b) 4/7; c) 1; 5. 91/216; 6. 7/8; 7. 0.172; 8. a) 0.5; b) 0.75; c) 0.167; 9. b) 2/3; c) 1/3;