Chapter 4

Chapter 4: ProbabilityChapter 4: Probability

ProbabilityProbability

• ProbabilityProbability is the chance of an event occurring.

• A probability experimentprobability experiment is a chance process that leads to well-defined results called outcomes.

• An outcomeoutcome is the result of a single trial of a probability experiment.

• A sample spacesample space is the set of all possible outcomes of a probability experiment.

ExampleExample

• Find the sample space for tossing two coins.

• Find the sample space for tossing a coin and rolling a die.

ExampleExample

• Use a tree diagram to determine the outcomes of an experiment of tossing three coins.

• An event is a set of outcomes. An event can be one outcome or more than one outcome.

• An event with one outcome is called a simple event.

• An event with more than one outcome is called a compound event.

ExampleExample

• The event of drawing a card and getting a queen of hearts is a _____________ event.

• The event of drawing a card and getting a spade is a ____________ event

• Classical probabilityClassical probability uses sample spaces to determine the numerical probability that an event will happen.

• Classical probability assumes that all outcomes in the sample space are equally likely to occur.

Formula for Classical Formula for Classical Probability – Probability –

• The probability of any event E is

_________________________________

_____ = ________

• Example: If a die is rolled one time, find these probabilities.

a) of getting a 4

• b) of getting an even number

• c) of getting a number greater than 4

• d) of getting a number greater than 3 and an odd number

Probability RulesProbability Rules

• The probability of an event E is a number between and including 0 and 1. 0 < P(E) < 1

• If an event E cannot occur, its probability is 0.

• If an event E is certain to occur its probability is 1.

• The sum of the probabilities of the outcomes in a sample space is 1.

Complementary EventsComplementary Events

• The complement of an event E is the set of outcomes in the sample space that are not included in the outcomes of event E. The complement of E is denoted by _______.

ExampleExample

• Find the complement of each event.• Flipping two coins and getting at

least one head

• Selecting a day of the week that has two syllables.

ExampleExample

• Find the complement of each event.– Rolling a die and getting a number

greater than 4

– Drawing a card and getting a face card

Rule for Complementary Rule for Complementary EventsEvents

• _______________ or

• __________________ or• • _______________

ExamplesExamples

• An urn contains three red marbles, eight white marbles and 3 green marbles. Find the probability of selecting a marble that is not green.

• Two dice are tossed. Find the probability of not getting doubles.

Classical vs. Empirical Classical vs. Empirical ProbabilityProbability

• The difference between classical and empirical probability is that classical probability assumes that certain outcomes are equally likely while empirical probability relies on actual observation to determine the likelihood of outcomes.

Formula for Empirical Formula for Empirical ProbabilityProbability

• Given a frequency distribution the probability of an event being in a given class is

_____= ___________________________

=_______• This probability is called empirical

probability and is based on observation.

Example:Example:• The director of the Readlot College Health

Center wishes to open an eye clinic. To justify the expense of such a clinic, the director reports the probability that a student selected at random from the college roster needs corrective lenses. He took a random sample of 500 students to compute this probability and found that 375 of them need corrective lenses. What is the probability that a Readlot College student selected at random needs corrective lenses?

ExampleExample

• The Right to Health Lobby wants to make a claim about the number of erroneous reports issued by a medical lab in one low-cost health center. Suppose they find in a random sample of 100 reports, 40 erroneous lab reports. What’s the probability that a report issued by this health center is erroneous?

Law of Large NumbersLaw of Large Numbers

• As the number of trials increases the empirical probability will approach the theoretical probability.

Subjective ProbabilitySubjective Probability

• Subjective probability uses a probability value based on an educated guess or estimate, employing opinions and inexact information.

ExampleExample

• Example: Classify each statement as an example of classical probability, empirical probability, or subjective probability.

• The probability that a person will watch the 6:00 news.

• The probability of winning at a chuck-a-luck game is 5/36.

ExampleExample

• An instructor states that the probability of passing the class, assuming that you pass the first test is 85%.

• The probability that a bus will be in an accident on a specific run is about 6%.

• The probability of getting a royal flush when five cards are selected at random is 1/649,740.

• The probability that a student will get a C or better in a statistics course is about 70%

• The probability that a new fast-food restaurant will be a success in Chicago is 35%.

• The probability that interest rates will rise in the next 6 months is 0.50.

The Addition Rule for The Addition Rule for ProbabilityProbability

• Two events are mutually exclusivemutually exclusive if they cannot occur at the same time (i.e. they have no outcomes in common).

Example: Mutual Example: Mutual ExclusivenessExclusiveness

• Determine whether these events are mutually exclusive:

• Roll a die and get an even number, and get a number less than 3

• Roll a die: Get a number greater than 3, and get a number less than 3

Example: Mutual Example: Mutual ExclusivenessExclusiveness

• Select a student in your college: The student is a sophomore, and the student is a business major.

• Select a registered voter: The voter is a Republican and the voter is a Democrat.

Addition Rule 1.Addition Rule 1.

• When two events A and B are mutually exclusive, the probability that A or B will occur is

•______________________________

Example:Example:

• An automobile dealer has 10 Fords, 7 Buicks, and 5 Plymouths on her used car lot. If a person purchases a used car, find the probability that it is a Ford or a Buick?

Example:Example:

• One card is randomly selected from a standard 52-card deck. Find the probability that the selected card is an ace or a king.

Example:Example:

• An automobile dealership has found that 37 percent of its new car sales have been dealer financed, 45 percent have been financed by another institution, and 18 percent have been cash sales. Find the probability that the next purchase of a new car at this dealership will be either a cash sale or dealer financed.

Addition Rule 2Addition Rule 2

• If A and B are not mutually exclusive, then

•___________________________

• Note: This rule can also be used when A and B are mutually exclusive since P(A and B) will always equal 0 for mutually exclusive events.

ExampleExample

• The probability that a student owns a car is 0.65, and the probability that a student owns a computer is 0.82. The probability that a student owns both is 0.55.

• What is the probability that a given student owns a car or a computer?

Example:Example:• The probability that a student owns a car

is 0.65, and the probability that a student owns a computer is 0.82. The probability that a student owns both is 0.55.

• What is the probability that a given student owns neither a car nor a computer?

ExampleExample

• A single card is drawn from a deck. Find the probability of selecting

• a four or a diamond

ExampleExample

• A jack or a black card

• a club or a diamond.

Example: TitanicExample: Titanic

• Example: Use the table below. Assume that one person aboard the Titanic is randomly selected.

• Find the probability of selecting a woman or a girl.

Titanic Mortality

Men Women Boys Girls Total

Survived

332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223


• Find the probability of selecting a woman or someone who survived.

Titanic Mortality


Survived

332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223


• Find the probability of selecting a woman or boy or girl.

Titanic Mortality


Survived

332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223


• Find the probability of selecting a woman or someone who died in the sinking of the ship.

Titanic Mortality


Survived

332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223


• Example: Use the table below. Assume that one person aboard the Titanic is randomly selected.

• Find the probability of selecting a woman or a girl.

Titanic Mortality


Survived

332 318 29 27 706

Died 1360 104 35 18 1517

Total 1692 422 64 45 2223

ExampleExample• At a used-book

sale, 100 books are adult books and 160 are children’s books. Seventy of the adult books are nonfiction while 60 of the children’s books are nonfiction. If a book is selected at random, find the probability that it is

• Fiction

• not a children’s nonfiction

• an adult book or a children’s nonfiction

4-4 The Multiplication Rules 4-4 The Multiplication Rules and Conditional Probabilityand Conditional Probability

• Two events A and B are independent events if the fact that A occurs does not affect the probability of B occurring.

• Flipping a coin and getting heads• Flipping a coin a second time and

getting heads• Speeding while driving to class• Getting a traffic ticket while driving to

class• Finding that your car will not start• Finding that your kitchen light will not

work

Multiplication Rule 1Multiplication Rule 1

• When two events are independent the probability of both occurring is

________________________

Example:Example:

• Find the probability of flipping a coin and getting tails and rolling a die and getting a 6.

Example:Example:

• One card is selected from a deck of 52 cards and replaced and then another card is selected. Find the probability of selecting a queen and then selecting a heart.

• Example: If 18% of all Americans are underweight, find the probability that if three Americans are selected at random, all will be underweight.

• The Multiplication Rule 1 can be extended to three or more independent events by using the formula

• ______________________________

Example:Example:

• The Gallup Poll reported that 52% of Americans used a seat belt the last time they got into a car. If four people are selected at random, find the probability that they all used a seat belt the last time they got into a car.

• When the outcome or occurrence of the first event affects the outcome or occurrence of the second event in such a way that the probability is changed, the events are said to be dependent events.

Conditional ProbabilityConditional Probability

• The conditional probabilityconditional probability of an event B in relationship to an event A is the probability that event B occurs after event A has already occurred. The notation for conditional probability is P(B|A).

Multiplication Rule 2Multiplication Rule 2

• When two events are dependent, the probability of both occurring is

P(A and B) = P(A) * P(B|A)

Example:Example:

• If two cards are selected from a standard deck of 52 cards without replacement, find these probabilities.

• Both are spades

Example:Example:


• Both are kings

Example:Example:


• Both are the same suit

ExampleExample

• A flashlight has six batteries, two of which are defective. If two are selected at random without replacement, find the probability that both are defective.

ExampleExample

• In a class containing twelve men and two women, 2 students are selected at random to given an impromptu speech. Find the probability that both are men.

Example:Example:

• An automobile manufacturer has three factories, A, B, and C. They produce 50%, 30%, and 20%, respectively of a specific model of car. Thirty percent of the cars produced in factory A are white, 40% of those produced in factory B are white, and 25% of those produced in factory C are white. If an automobile produced by the company is selected at random, find the probability that it is white.

Conditional ProbabilityConditional Probability

• Conditional Probability – Derived from formula for dependent events

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

• The formula is __________________.

Example:Example:

• At a small college, the probability that a student takes physics and sociology is 0.092. The probability that a student takes sociology is 0.73. Find the probability that the student is taking physics, given that he or she is taking sociology.

ExampleExample

• A circuit to run a model railroad has eight switches. Two are defective. If a person selects two switches at random and tests them, find the probability that the second one is defective, given that the first one is defective.

ExampleExample

• In a pizza restaurant, 95% of the customers order pizza. If 65%of the customers order pizza and a salad, find the probability that a customer who orders pizza will also order a salad.

ExampleExample

• The probability that it snows and the bus arrives late is 0.023. John hears the weather forecast, and there is a 40% chance of snow tomorrow. Find the probability that the bus will be late, given that it snows.

ExampleExample

• Try at Home for Next Time - Thirteen percent of the employees of a large company are female technicians. Forty percent of its workers are technicians. If a technician has been assigned to a particular job, what is the probability that the person is female?

• Example: The medal distribution from the 2000 Summer Olympic Games is shown in the table.

• Find the probability that the winner won the gold medal, given that the winner was from the United States.

Gold Silver Bronze

United States 39 25 33

Russia 32 28 28

China 28 16 15

Australia 16 25 17

Others 186 205 235

• Example: The medal distribution from the 2000 Summer Olympic Games is shown in the table.

• Find the probability that the winner was from the United States, given that he or she won the gold medal.

Gold Silver Bronze

United States 39 25 33

Russia 32 28 28

China 28 16 15

Australia 16 25 17

Others 186 205 235

Example:Example:• Traffic entering an intersection can

continue straight ahead or turn right. Eighty percent of the traffic flow is straight ahead. If a car continues straight, the probability of a collision is 0.0004; if a car turns right, the probability of a collision is 0.0036. Find the probability that a car entering the intersection will have a collision.

Example:Example:

• In situations where it is critical that a system function properly, additional backup systems are usually provided. Suppose a switch is used to activate a component in a satellite. If the switch fails, then a second switch takes over and activates the component. If each switch has a probability of 0.002 of failing, what is the probability that the component will be activated?

ExampleExample• A vaccine has a 90% probability of being

effective in preventing a certain disease. The probability of getting the disease if a person is not vaccinated is 50%. In a certain geographic region, 25% of the people get vaccinated. If a person is selected at random, find the probability that he or she will contract the disease.

At Least OneAt Least One

• The complement of at least one is zero of the same type.

Example:Example:

• 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.

ExampleExample

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

ExampleExample

• It has been found that 40% of all people over the age of 85 suffer from Alzheimer’s disease. If three people over 85 are selected at random, find the probability that at least one person does not suffer from Alzheimer’s disease.

Example:Example:

• Among a class of 25 students, find the probability that at least two of them have the same birthday.

Example:Example:

• In a lab there are eight technicians. Three are male and five are female. If three technicians are selected, find the probability that at least one is female.

Example:Example:

• On a surprise quiz consisting of five true-false questions, an unprepared student guesses each answer. Find the probability that he gets at least one correct.

Example:Example:

• A medication is 75% effective against a bacterial infection. Find the probability that if 12 people take the medications, at least one person’s infection will not improve.

Fundamental Counting Fundamental Counting RuleRule

• In a sequence of n events in which the first one has k possibilities and the second has k2 possibilities and the third has k3 possiblities and so fourth, the total number of possibilities of the sequence would be k1* k2 * k3. . . .

ExampleExample

• There are eight different statistics books, 6 different geometry books and 3 different trigonometry books. A student must select one book of each type. How many different ways can this be done?

Example:Example:

• A college bookstore offers a personal computer system consisting of a computer, a monitor, and a printer. A student has a choice of two computers, three monitors, and two printers, all of which are compatible. In how many ways can a computer system be bundled?

• Factorial Formulas – For any counting number, n,

• n! = _________________

• 0! = _________________

Factorial Rule Factorial Rule

• The arrangement of n objects is ________. (This is a special case of the permutation rule, where all n objects are being arranged.)

Example:Example:

• You are hosting a dinner party for eight people. In preparing the seating arrangement, you would like to know the number of different ways in which the guest can be arranged.

Example:Example:

• Five students are to give presentations in class on a particular day. In how many ways can the presentations be made?

PermutationPermutation

• A permutation is an arrangement of n objects selected r at a time in a specific order.

ExampleExample

• How many ordered seating arrangements can be made for eight people in five chairs?

• The board of directors of a local college has 12 members. Three officers—president, vice-president and treasurer—must be elected from the members. How many different possible slates of officers are there?

CombinationCombination

• A selection of distinct objects without regard to order is called a combination.

Combination RuleCombination Rule

• The number of combinations of r objects selected from n objects is denoted by _________ and is given by the formula

• ______________

Example:Example:

• How many different 5-card poker hands can be dealt from a standard deck of 52 cards?

Example:Example:

• A professor grades homework by randomly choosing 5 out of 12 homework problems to grade. How many different groups of problems can he possibly grade?

Example:Example:

• A sales representative must visit four cities: Omaha, Dallas, Wichita, and Oklahoma City. There are air connections between each of the cities. In how many orders can he visit the cities?

ExampleExample

• A pizza shop offers a combination pizza consisting of a choice of any three of the four ingredients: pepperoni (P), mushrooms (M), sausage (S), and anchovies (A). Determine the number of possible combination pizzas.

ExampleExample

• A professor grades homework by randomly choosing 5 out of 12 homework problems to grade. How many different groups of problems can he possibly grade?

Example:Example:

• There are three nursing positions to be filled at Lilly Hospital. Position one is the day nursing supervisor; position two is the night nursing supervisor; and position three is the nursing coordinator position. There are 15 candidates qualified for all three of the positions. In how many ways can the positions be filled by the applicants?

Example:Example:

• A parent-teacher committee consisting of 4 people is to be formed from 20 parents and 5 teachers. Find the probability that the committee will consist of these people. (Assume that the selection will be random.)

• All teachers

Example:Example:


• 2 teachers and 2 parents

Example:Example:


• All parents

Example:Example:


• 1 teacher

Example:Example:

• An instructor gives her class a quiz that consists of 2 true/false questions, one multiple choice question with five selections, and 2 multiple choice questions with 4 choices each. One student did not prepare for the quiz and decides to randomly guess. In how many ways can the student fill out the quiz?

• What’s the probability that the student gets a score of 100?

Example:Example:

• An instructor give her class a quiz that consists of 2 true/false questions, one multiple choice question with five selections, and 2 multiple choice questions with 4 choices each. One student did not prepare for the quiz and decides to randomly guess. In how many ways can the student fill out the quiz?

• What’s the probability that the student gets a score of 80?

Example:Example:

• An insurance sales representative selects three policies to review. The group of policies she can select from contains 8 life policies, 5 automobile polices, and 2 homeowner’s policies. Find the probability of selecting:

• All life policies

Example:Example:


• Both homeowner’s policies

Example:Example:


• All automobile policies

Example:Example:


• 1 of each policy.

Technology

Chapter 4