Ch5 Probability

______is a measure of the likelihood of a random phenomenon or chance behavior. Probability describes the long-term proportion with which a certain outcome will occur in situations with short-term uncertainty.

If we flip a coin 100 times and compute the proportion of heads observed after each toss of the coin, what will the proportion approach?

The Law of Large Numbers As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome.

Ch 5.1 Probability Rule Objective A : Sample Spaces and Events

Experiment – any activity that leads to well-defined results called outcomes.

Outcome – the result of a single trial of a probability experiment.

Sample space, S – the set of all possible outcomes of a probability experiment.

Event, E – a subset of the sample space

Simple event, ei – an event with one outcome is called a simple event.

Compound event – consists of two or more outcomes.

Example 1: A die is tossed one time. (a) List the elements of the sample space S.

(b) List the elements of the event consisting of a number that is greater than 4.

Example 2: A coin is tossed twice. List the elements of the sample space S, and list the elements of the event consisting of at least one head.

Objective B : Requirements for Probabilities

1. Each probability must lie on between 0 and 1. (0PE ( ) 1) 2. The sum of the probabilities for all simple events in S equals 1.

(Pe (i ) 1)

If an event is impossible, the probability of the event is 0. If an event is a certainty, the probability of the event is 1. An unusual event is an event that has a low probability of occurring. Typically, an event with a probability less than 0.05 is considered as unusual.

Probabilities should be expressed as reduced fractions or rounded to three decimal places.

Example 1: A probability experiment is conducted. Which of these can be considered a probability of an outcome?

(a) 2/5 (b) -0.28 (c) 1.09

Example 2: Why is the following not a probability model?

Color Probability Red 0.28 Green 0.56 Yellow 0.37

Example 3: Given: 푆 = {푒1, 푒2, 푒3, 푒4} 푃(푒1) = 푃(푒2) = 0.2 and 푃(푒3) = 0.5 Find: 푃(푒4)

Objective C : Calculating Probabilities

C1. Approximating Probabilities Using the Empirical Approach (Relative Frequency Approximation of Probability)

The probability of event E is approximately the number of times event is observed divided by the number of repetitions of the experiment.

Frequency of E PEE( ) RelativeFrequencyof Number of Trials of Experiement

Example 1: Suppose that you roll a die 100 times and get six 80 times. Based on these results, what is the probability that the next roll results in six?

Example 2: During a sale at men’s store, 16 white sweaters, 3 red sweaters, 9 blue sweaters, and 7 yellow sweaters were purchased. If a customer is selected at random, find the probability that he bought a sweater that was not white.

Example 3: The age distribution of employees for this college is shown below:

Age Number of Employees Under 20 25 20 – 29 48 30 – 39 32 40 – 49 15 50 and over 10

If an employee is selected at random, find the probability that he or she is in the following age groups (a) Between 30 and 39 years of age

(b) Under 20 or over 49 years of age

C2. Classical Approach to Probability (Equally Likely Outcomes are required)

If an experiment has n equally likely outcomes and if the number of ways that an event E can occur in m , then the probability of E , PE(), is

Number of ways that Em can occur PE() Number of possible outcomes n

If S is the sample space of this experiment, NE() PE() NS() where NE()is the number of outcomes in event E , and NS()is the number of outcomes in the sample space.

Example 1: Let the sample space be S 1,2,3,4,5,6,7,8,9,10. Suppose the outcomes are equally likely.

(a) Compute the probability of the event F  5,9 .

(b) Compute the probability of the event E = "an odd number."

Example 2: Two dice are tossed. Find the probability that the sum of two dice is greater than 8?

Example 3: If one card is drawn from a deck, find the probability of getting (a) a club; (b) a 4 and a club

Example 4: Three equally qualified runners, Mark, Bill, and Alan, run a 100-meter sprint, and the order of finish is recorded. (a) Give a sample space S . (b) What is the probability that Mark will finish last?

Ch 5.2 The Addition Rules and Complements

Objective A : Addition Rule for Disjoint (Mutually Exclusive) Events

Event A and B are disjoint (mutually exclusive) if they have no outcomes in common.

Addition Rule for Disjoint Events

If E and F are disjoint events, then PEFPEPF( or ) ( ) ( ) .

Example 1. A standard deck of cards contains 52 cards. One card is randomly selected from the deck. Compute the probability of randomly selecting a two or three from a deck of cards.

Objective B : General Addition Rule

The General Addition Rule

For any two events and , PEFPEPFPEF( or ) ( ) ( ) ( and ).

Example 1 : A standard deck of cards contains 52 cards. One card is randomly selected from the deck. Compute the probability of randomly selecting a two or club from a deck of cards.

Objective C : Complement Rule

Complement Rule If E represents any event and Ec represents the complement of , then PEPE(C ) 1 ( )

e.g. The chance of raining tomorrow is 70%. What is the probability that it will not rain tomorrow?

Example 1: A probability experiment is conducted in which the sample space of the experiment is S 1,2,3,4,5,6,7,8,9,10,11,12. Let event E 2,3,5,6,7, event F  5,6,7,8 , and event G  9,11

(a) List the outcome in EF and . Are mutually exclusive?

(b) Are FG and mutually exclusive? Explain.

(d) Determine using the General Addition Rule.

(e) List the outcomes in EC . Find PE()C by counting the number of outcomes in .

(f) Determine PE()C using the Complement Rule.

Example 2: In a large department store, there are 2 managers, 4 department heads, 16 clerks, and 4 stock persons. If a person is selected at random, (a) find the probability that the person is a clerk or a manager; (b) find the probability that the person is not a clerk.

Example 3: The following probability shows the distribution for the number of rooms in U.S. housing units.

(a) Verify that this is a probability model.

(b) What is the probability that a randomly selected housing unit has four or more rooms? Interpret this probability.

Example 4: According to the U.S. Census Bureau, the probability that a randomly selected household speaks only English at home is 0.81. The probability that a randomly selected household speaks only Spanish at home is 0.12.

(a) What is the probability that a randomly selected household speaks only English or only Spanish at home?

(b) What is the probability that a randomly selected household speaks a language other than only English at home?

Objective D : Contingency Table

A contingency table relates two categories of data. It is also called a two-way table which consists of a row variable and a column variable. Each box inside the table is called a cell.

Example 1: In a certain geographic region, newspapers are classified as being published daily morning, daily evening, and weekly. Some have a comics section and other do not. The distribution is shown here.

Have comics Section Morning Evening Weekly

Yes 2 3 1

No 3 4 2

If a newspaper is selected at random, find these probabilities.

(a) The newspaper is a weekly publication.

(b) The newspaper is a daily morning publication or has comics.

Ch 5.3 Independence and the Multiplication Rule

Objective A : Independent Events

Two events are independent if the occurrence of event E does not affect the probability of event F . Two events are dependent if the occurrence of event affects the probability of event .

Example 1: Determine whether the events E and F are independent or dependent. Justify your answer.

(a) : The battery in your cell phone is dead. : The battery in your calculator is dead.

(b) : You are late to class. : Your car runs out of gas.

Objective B : Multiplication Rule for Independent Events

If E and F are independent events, then PEFPEPF( and ) ( ) ( )

Example 1: If 36% of college students are underweight, find the probability that if three college students are selected at random, all will be underweight.

Example 2: If 25% of U.S. federal prison inmates are not U.S. citizens, find the probability that two randomly selected federal prison inmates will be U.S. citizens.

Objective C : At-Least Probabilities

Probabilities involving the phrase “at least” typically use the Complement Rule. The phrase at least means “greater than or equal to.” For example, a person must be at least 17 years old to see an R-rated movie.

Example 1: If you make random guesses for four multiple-choice test questions (each with five possible answers), what is the probability of getting at least one correct?

Example 2: For the fiscal year 2007, the IRS audited 1.77% of individual tax returns with income of $100,000 or more. Suppose this percentage stays the same for the current fiscal year.

(a) Would it be unusual for a return with income of $100,000 or more to be audited?

(b) What is the probability that two randomly selected returns with income of $100,000 or more will be audited?

(d) What is the probability that at least one of the two randomly selected returns with income $100,000 or more will be audited?

Ch 5.4 Conditional Probability and the General Multiplication Rule

Objective A : Conditional Probability and the General Multiplication Rule

A1. Multiplication Rule for Dependent Events

If E and F are dependent events, then PEFPEPFE( and ) ( ) ( | ) .

The probability of and is the probability of event occurring times the probability of event occurring, given the occurrence of event .

Example 1: A box has 5 red balls and 2 white balls. If two balls are randomly selected (one after the other), what is the probability that they both are red?

(a) With replacement

(b) Without replacement

Example 2: Three cards are drawn from a deck without replacement. Find the probability that all are jacks.

A2. Conditional Probability

PEFNEF( and ) ( and ) If E and F are any two events, then PFE(). PENE()()

The probability of event F occurring, given the occurrence of event E , is found by dividing the probability of E and F by the probability of E . 12

Example 1: At a local Country Club, 65% of the members play bridge and swim, and 72% play bridge. If a member is selected at random, find the probability that the member swims, given that the member plays bridge.

B. Application

Example 1: Eighty students in a school cafeteria were asked if they favored a ban on smoking in the cafeteria. The results of the survey are shown in the table.

Class Favor Oppose No opinion

Freshman 15 27 8

Sophomore 23 5 2

If a student is selected at random, find these probabilities.

(a) The student is a freshman or favors the ban.

(b) Given that the student favors the ban, the student is a sophomore.

Example 2 : The local golf store sells an “onion bag” that contains 35 “experienced” golf balls. Suppose that the bag contains 20 Titleists, 8 Maxflis and 7 Top-Flites. 13

(a) What is the probability that two randomly selected golf balls are both Titleists?

(b) What is the probability that the first ball selected is a Titleist and the second is a Maxfli?

(d) What is the probability that one golf ball is a Titleist and the other is a Maxfli?

Ch 5.5 Counting Techniques

Objective A : Multiplication Rule of Counting

If Task 1 can be done in n ways and Task 2 can be done in m ways, Task 1 and Task 2 performed together can be done in nm ways.

Example 1: Two dice are tossed. How many outcomes are in the sample space?

Example 2: A password consists of two letters followed by one digit. How many different passwords can be created? (Note: Repetitions are allowed)

Objective B : Counting Techniques : Permutation and Combination

Permutation The number of ways we can arrange n distinct objects, taking them r at a time, is n! P  Order Matters nr ()!nr

Combination The number of distinct combinations of distinct objects that can be formed, taking them at a time, is

n! C  Order doesn’t matter nr r!( n r )!

Example: Given the letters A, B, C, and D, list the permutations and combinations for selecting two letters.

Example 1: Find (a) 5! (b) 25P 8 (c) 12P 4 (d) 25C 8

Example 2: An inspector must select 3 tests to perform in a certain order on a manufactured part. He has a choice of 7 tests. How many ways can be performed 3 different tests?

Example 3: If a person can select 3 presents from 10 presents, how many different combinations are there?

Objective C: Using the Counting Techniques to Find Probabilities

After using the multiplication rule, combination, and permutation learned from this section to count the number of outcomes for a sample space, NS(), and the number of outcomes for an NE() event, NE(), we can calculate PE()by the formula PE() NS()

Example 1: A Social Security number is used to identify each resident of the United States uniquely. The number is of the form xxx-xx-xxxx, where each x is a digit from 0 to 9.

(a) How many Social Security numbers can be formed?

(b) What is the probability of correctly guessing the Social Security number of the President of the United States?

Example 2 : Suppose that there are 55 Democrats and 45 Republicans in the U.S. Senate. A committee of seven senators is to be formed by selecting members of the Senate randomly. (a) What is the probability that the committee is composed of all Democrats?

(b) What is the probability that the committee is composed of all Republicans?

Example 3 : Five cards are selected from a 52-card deck for a poker hand. (A poker hand consists of 5 cards dealt in any order.)

(a) How many outcomes are in the sample space?

(b) A royal flush is a hand that contains that A, K, Q, J, 10, all in the same suit. How many ways are there to get a royal flush?

5.6 Putting It Together: Which Method Do I Use? Objectives Determine the appropriate probability rule to use Determine the appropriate counting technique to use