Section 4-4 Multiplication Rule: Basics

Key Concept

The basic multiplication rule is used for finding P(A and B), the probability that event A occurs in a first trial and event B occurs in a second trial.

If the outcome of the first event A somehow affects the probability of the second event B, it is important to adjust the probability of B to reflect the occurrence of event A.

NotationP(A and B) = P(event A occurs in a first trial and event B occurs in a second trial)

Tree Diagrams

A tree diagram is a picture of the possible outcomes of a procedure, shown as line segments emanating from one starting point. These diagrams are sometimes helpful in determining the number of possible outcomes in a sample space, if the number of possibilities is not too large.

Tree Diagrams

This figure summarizes the possible outcomes for a true/false question followed by a multiple choice question.

Note that there are 10 possible combinations.

Example 1: Create a tree diagram for first tossing a coin and then rolling a die.

Heads

Tails

Conditional ProbabilityKey Point

We must adjust the probability of the second event to reflect the outcome of the first event.

Conditional ProbabilityImportant Principle

The probability for the second event B should take into account the fact that the first event A has already occurred.

Notation for Conditional Probability

P(B | A) represents the probability of event B occurring after it is assumed that event A has already occurred (read B | A as “B given A.”)

Dependent and Independent

Two events A and B are independent if the occurrence of one does not affect the probability of the occurrence of the other. (Several events are similarly independent if the occurrence of any does not affect the probabilities of the occurrence of the others.) If A and B are not independent, they are said to be dependent.

Dependent Events

Two events are dependent if the occurrence of one of them affects the probability of the occurrence of the other, but this does not necessarily mean that one of the events is a cause of the other.

Treating Dependent Events as Independent

Some calculations are cumbersome, but they can be made manageable by using the common practice of treating events as independent when small samples are drawn from large populations. In such cases, it is rare to select the same item twice.

The 5% Guideline for Cumbersome Calculations

If a sample size is no more than 5% of the size of the population, treat the selections as being independent (even if the selections are made without replacement, so they are technically dependent).

Example 2: For the given pair of events, classify them as independent or dependent. If the two events are technically dependent but can be treated as if they are independent according to the 5% guideline, consider them to be independent.

Randomly selecting a TV viewer who is watching Saturday Night Live.

Randomly selecting a second TV viewer who is watching Saturday Night Live.

Independent, since the population is so large that the 5% guideline applies.

Example 3: For the given pair of events, classify them as independent or dependent. If the two events are technically dependent but can be treated as if they are independent according to the 5% guideline, consider them to be independent.

Wearing plaid shorts with black socks and sandals.

Asking someone on a date and getting a positive response.

Dependent, if there is visual contact during the invitation, since visual impressions affect responses. Independent, if there is no visual contact during the invitation, since the invitee would have no knowledge of the inviter’s clothing.

Formal Multiplication Rule

*P(A and B) = P(A) • P(B | A)

*Note that if A and B are independent events, P(B | A) is really the same as P(B).

Intuitive Multiplication Rule

When finding the probability that event A occurs in one trial and event B occurs in the next trial, multiply the probability of event A by the probability of event B, but be sure that the probability of event B takes into account the previous occurrence of event A.

Applying the Multiplication Rule

Be sure to find the probability of event B assuming that event A has

already occurred.

Caution

When applying the multiplication rule, always consider whether the events are independent or dependent, and adjust the calculations accordingly.

Example 4: Use the data in the following table, which summarizes blood groups and Rh types for 100 subjects. These values may vary in different regions according to the ethnicity of the population.

If 2 of the 100 subjects are randomly selected, find the probability that they are both group O and type Rh+.

a) Assume that the selections are made with replacement.

Group

Type Rh+ 39 35 8 4

Rh– 6 5 2 1

39O and Rh+

100P

39 390.152

100 100

Example 4: Use the data in the following table, which summarizes blood groups and Rh types for 100 subjects. These values may vary in different regions according to the ethnicity of the population.

If 2 of the 100 subjects are randomly selected, find the probability that they are both group O and type Rh+.

b) Assume that the selections are made without replacement.

Group

Type Rh+ 39 35 8 4

Rh– 6 5 2 1

39O and Rh+

100P

39 380.150

100 99

Multiplication Rule for Several Events

In general, the probability of any sequence of independent events is simply the product of their corresponding probabilities.

Example 5: Use the data in the following table, which summarizes blood groups and Rh types for 100 subjects. These values may vary in different regions according to the ethnicity of the population.

People with blood that is group O and type Rh– are considered to be universal donors, because they can give blood to anyone. If 4 of the 100 subjects are randomly selected, find the probability that they are all universal donors.

a) Assume that the selections are made with replacement.

Group

Type Rh+ 39 35 8 4

Rh– 6 5 2 1

6O and Rh

100P

6 6 6 60.000013

100 100 100 100

Example 5: Use the data in the following table, which summarizes blood groups and Rh types for 100 subjects. These values may vary in different regions according to the ethnicity of the population.

People with blood that is group O and type Rh– are considered to be universal donors, because they can give blood to anyone. If 4 of the 100 subjects are randomly selected, find the probability that they are all universal donors.

b) Assume that the selections are made without replacement.

Group

Type Rh+ 39 35 8 4

Rh– 6 5 2 1

6O and Rh

100P

6 5 4 30.00000383

100 99 98 97

Principle of Redundancy

One design feature contributing to reliability is the use of redundancy, whereby critical components are duplicated so that if one fails, the other will work. For example, single-engine aircraft now have two independent electrical systems so that if one electrical system fails, the other can continue to work so that the engine does not fail.

Summary of Fundamentals*In the addition rule, the word “or” in P(A or B) suggests addition. Add P(A) and P(B), being careful to add in such a way that every outcome is counted only once.

*In the multiplication rule, the word “and” in P(A and B) suggests multiplication. Multiply P(A) and P(B), but be sure that the probability of event B takes into account the previous occurrence of event A.

Example 6: A quick quiz consists of a true/false question followed by a multiple-choice question with four possible answers (a, b, c, d). An unprepared student makes random guesses for both answers.

a) Consider the event of being correct with the first guess and the event of being correct with the second guess. Are those two events independent?

Yes, since the corrections of the first response have no effect on the correctness of the second, the events are independent.

b) What is the probability that both answers are correct?

1 2 1 2 1C and C C C | C

1 1

2 4

0.125

P P P

Example 6: A quick quiz consists of a true/false question followed by a multiple-choice question with four possible answers (a, b, c, d). An unprepared student makes random guesses for both answers.

c) Based on the results, does guessing appear to be a good strategy?

No, since 0.125 > 0.05, getting both answers correct would not be unusual – but it certainly isn’t likely enough to recommend such a practice.

Example 7: The Wheeling Tire Company produced a batch of 5000 tires that includes exactly 200 that are defective.

a) If 4 tires are randomly selected for installation on a car, what is the probability that they are all good?

Let G = getting a tire that is good.

48000.96

5000P G

1 2 3 4 1 2 1 3 1 2 3 1 2 3 and and and | | and | and and

4800 4799 4798 4797

5000 4999 4998 4997

0.849

P G G G G P G P G G P G G G P G G G G

Example 7: The Wheeling Tire Company produced a batch of 5000 tires that includes exactly 200 that are defective.

b) If 100 tires are randomly selected for shipment to an outlet, what is the probability that they are all good? Should this outlet plan to deal with defective tires returned by consumers?

Since n = 100 represents 100/5000 = 0.02 ≤ 0.05 of the population, use the 5% guideline and treat the repeated selections as being independent.

Yes; since 0.0169 ≤ 0.05, getting 100 good tires would be an unusual and the outlet should plan on dealing with returns of defective tires.

1 2 100 1 2 100

100

and ...and ...

0.96 0.96 ... 0.96

0.96

0.0169

P G G G P G P G P G

Example 8: The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume that your alarm clock has a 0.9 probability of working on any given morning.

a) What is the probability that your alarm clock will not work on the morning of an important exam?

Let A = the alarm works.P(A) = 0.9 for each alarm

b) If you have two such alarm clocks, what is the probability that they both fail on the morning of an important exam?

1 ( ) 1 0.9 0.1P A P A

1 2 1 2 and

0.1 0.1

0.01

P A A P A P A

Example 8 continued: The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume that your alarm clock has a 0.9 probability of working on any given morning.

c) With one alarm clock, you have a 0.9 probability of being awakened. What is the probability of being awakened if you have two alarm clocks?

Note: Parts (b) and (c) assume that the alarm clocks work independently of each other. This would not be true if they are both electric alarm clocks.

1 2

1 2

being awakened and

1 and

1 0.01

0.99

P P A A

P A A

Example 8 continued: The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume that your alarm clock has a 0.9 probability of working on any given morning.

d) Does a second alarm clock result in greatly improved reliability?

Yes, the probability of not being awakened drops from 1 chance in 10 to one chance in 100.