Sec 4.4 The multiplication Rule and conditional probability.

Sec 4.4

The multiplication Rule and conditional probability

Independent events

Two events A and B are independent events if the fact that A occurs does NOT effect the probability of B occurring.

Multiplication Rule 1

If two events are independent, the probability of both happening is

P(A and B)= P(A) • P(B)

Examples

A coin is flipped and a die is rolled at the same time. What is the probability of getting a tail and a 4?

A card is drawn at random and then replaced. What is the probability of drawing a queen first and then an ace?

More examples

A box contains 3 red balls, 2 blue balls and 5 green balls. A ball is selected, replaced and a second ball is selected.

Write the sample space.

Write the probability of each event.

Refer to the last example and answer the following probabilities:

a) Selecting 2 blue balls.

b) Selecting 1 blue ball and then 1 green ball.

c) Selecting 1 red ball and then 1 blue.

Color blindness

Studies show that approximately 9% of men have color blindness. If 3 males are selected at random what is the probability that none of them have color blindness?

Refer to the color blindness problem

If 3 men are chosen at random. What is the sample space for color blindness?

Do you think each event has the same probability of happening?

Dependent events

When the outcome of the first event affects the outcome of the second event in such a way that the probability is changed, the events are dependent.

Multiplication Rule #2

When two events are dependent, the probability of both happening is

P(A and B)= P(A) · P(B|A)

Example 4-28

A person owns a collection of 30 CDs, of which 5 are country music. If 2 CDs are selected at random, find the probability that both are country music.

Example 4-29

The World Wide Insurance Company found that 53% of the residents of a city had homeowner’s insurance (H) with the company. Of these clients, 27% also had automobile insurance (A) with the company. If a resident is selected at random, find the probability that the resident has both homeowner’s and automobile insurance with the World Wide Insurance Company.

Example 4-30

Three cards are drawn from an ordinary deck and not replaced. Find the probability of these.

A.) Getting 3 jacks. B.) Getting an ace, a king, and a queen in order. C.) Getting a club, a spade, and a heart in order. D.) Getting 3 clubs.

Example 4-31

Box 1 contains 2 red balls and 1 blue ball. Box 2 contains 3 blue balls and 1 red ball. A coin is tossed. If it falls heads up, box 1 is selected and a ball is drawn. If it falls tails up, box 2 is selected and a ball is drawn. Find the probability of selecting a red ball.

Formula for Conditional Probability

The probability that the second event B occurs given that the first event A has occurred can be found by dividing the probability that both events occurred by the probability that the first event has occurred. The formula is:

)(

)()(

Ap

AandBpABp

Example 4-32

A box contains black chips and white chips. A person selects two chips without replacement. If the probability of selecting a black chip and a white chip is 15/56, and the probability of selecting a black chip on the first draw is 3/8, find the probability of selecting the white chip on the second draw, given that the first chip selected was a black chip.

Example 4-33

The probability that Sam parks in a no-parking zone and gets a parking ticket is 0.06, and the probability that Sam cannot find a legal parking space and has to park in the no-parking zone is 0.20. On Tuesday, Sam arrives at school and has to park in a no-park zone. Find the probability that he will get a parking ticket.

Example 4-34

A recent survey asked 100 people if they thought women in the armed forces should be permitted to participate in combat. The results of the survey are shown.

Male Yes-32, No-18 Total-50 Female Yes-8 No-42 Total-50 Find these probabilities. A.) The respondent answered yes, given that the

respondent was a female. B.) The respondent was a male, given that the respondent

answered no.

Probabilities for “At Least”

The multiplication rules can be used with the complementary event rule (Section 4-2) to simplify solving probability problems involving “at least.” Examples 4-35, 4-36, and 4-37 illustrate how this is done.

Example 4-35

A game is played by drawing four cards from an ordinary deck and replacing each card after it is drawn. Find the probability of winning if at least one ace is drawn.

Example 4-36

A coin is tossed 5 times. Find the probability of getting at least one tail.

Example 4-37

The Newckware Association of America reported that 3% of ties sold in the United States are bow ties. If 4 customers who purchased a tie are randomly selected, find the probability that at least one purchased a bow tie.

Homework

Sec 4.4 page 209-212#1-51 Every other odds. i.e 1,5,9,

etc.