Larson/Farber Ch. 4

Elementary StatisticsLarson Farber

4

x = number of on time arrivals

x = number of points scored in a game

x = number of employees reaching sales quota

x = number of correct answers

Discrete Probability Distributions

Larson/Farber Ch. 4

Probability Distributions

Section 4.1

Larson/Farber Ch. 4

A random variable, x is the numerical outcome of a probability experiment.

x = The number of people in a car

x = The gallons of gas bought in a week

x = The time it takes to drive from home to school

x = The number of trips to school you make per week

Random Variables

Larson/Farber Ch. 4

A random variable is discrete if the number of possible outcomes is finite or countable. Discrete random variables are determined by a count.

A random variable is continuous if it can take on any value within an interval. The possible outcomes cannot be listed. Continuous random variables are determined by a measure.

Types of Random Variables

Larson/Farber Ch. 4

x = The number of people in a car

x = The gallons of gas bought in a week

x = The time it takes to drive from home to school

x = The number of trips to school you make per week

Identify each random variable as discrete or continuous.

Discrete – you count the number of people in a car 0, 1, 2, 3… Possible values can be listed.

Continuous – you measure the gallons of gas. You cannot list the possible values.

Continuous – you measure the amount of time. The possible values cannot be listed.

Discrete – you count the number of trips you make. The possible numbers can be listed.

Types of Random Variables

Larson/Farber Ch. 4

A discrete probability distribution lists each possible value of the random variable, together with its probability.

A survey asks a sample of families how many vehicles each owns.

x P(x)0 0.0041 0.4352 0.3553 0.206

number ofvehicles

Properties of a probability distribution• Each probability must be between 0 and 1, inclusive.

• The sum of all probabilities is 1.

Discrete Probability Distributions

Larson/Farber Ch. 4

• The height of each bar corresponds to the probability of x.

• When the width of the bar is 1, the area of each bar corresponds to the probability the value of x will occur.

0 1 2 3

0

.10

.20

.30

.40

P(x

)

0.004

0.435

0.355

0.206

Number of Vehicles

x0 1 2 3

Probability Histogram

Larson/Farber Ch. 4

Mean, Variance and Standard Deviation

The variance of a discrete probability distribution is:

The standard deviation of a discrete probability distribution is:

The mean of a discrete probability distribution is:

Larson/Farber Ch. 4

Mean (Expected Value)

Multiply each value by its probability. Add the products

The expected value (the mean) is 1.763 vehicles.

x P(x) xP(x)0 0.004 01 0.435 0.4352 0.355 0.713 0.206 0.618

1.763

Calculate the mean

Larson/Farber Ch. 4

Calculate the Variance and Standard Deviation

The standard deviation is 0.775 vehicles.

The mean is 1.763 vehicles.

variance

x P(x) x- (x - ) P(x)(x - )0 0.004 -1.763 3.108 0.0121 0.435 -0.763 0.582 0.2532 0.355 0.237 0.056 0.0203 0.206 1.237 1.530 0.315

0.601

μ μ P(x)

Larson/Farber Ch. 4

Binomial Distributions

Section 4.2

Larson/Farber Ch. 4

• There are a fixed number of trials. (n)• The n trials are independent and repeated under identical

conditions.• Each trial has 2 outcomes, S = Success or F = Failure.• The probability of success on a single trial is p. P(S) = p The probability of failure is q. P(F) =q where p + q = 1• The central problem is to find the probability of x

successes out of n trials. Where x = 0 or 1 or 2 … n.

Binomial ExperimentsCharacteristics of a Binomial Experiment

The random variable x is a count of the number of successes in n trials.

Larson/Farber Ch. 4

1. What is the 11th digit after the decimal point for the irrational number e?(a) 2 (b) 7 (c) 4 (d) 5

2. What was the Dow Jones Average on February 27, 1993?(a) 3265 (b) 3174 (c) 3285 (d) 3327

3. How many students from Sri Lanka studied at U.S. universities from 1990-91?(a) 2320 (b) 2350 (c) 2360 (d) 2240

4. How many kidney transplants were performed in 1991?(a) 2946 (b) 8972 (c) 9943 (d) 7341

5. How many words are in the American Heritage Dictionary?(a) 60,000 (b) 80,000 (c) 75,000 (d) 83,000

Guess the Answers

Larson/Farber Ch. 4

Quiz Results

Count the number of correct answers. Let the number of correct answers = x.

Why is this a binomial experiment?

What are the values of n, p and q?

What are the possible values for x?

The correct answers to the quiz are:1. d 2. a 3. b 4. c 5. b

Larson/Farber Ch. 4

A multiple choice test has 8 questions each of which has 3 choices, one of which is correct. You want to know the probability that you guess exactly 5 questions correctly.Find n, p, q, and x.

A doctor tells you that 80% of the time a certain type of surgery is successful. If this surgery is performed 7 times, find the probability exactly 6 surgeries will be successful. Find n, p, q, and x.

n = 8 p = 1/3 q = 2/3 x = 5

n = 7 p = 0.80 q = 0.20 x = 6

Binomial Experiments

Larson/Farber Ch. 4

Find the probability of getting exactly 3 questions correct on the quiz.

Write the first 3 correct and the last 2 wrong as SSSFFP(SSSFF) = (.25)(.25)(.25)(.75)(.75) = (.25)3(.75)2 = 0.00879

Since order does not matter, you could get any combination of three correct out of five questions. List these combinations.

SSSFF SSFSF SSFFS SFFSS SFSFSFFSSS FSFSS FSSFS SFSSF FFSSF

Each of these 10 ways has a probability of 0.00879.

P(x = 3) = 10(0.25)3(0.75)2 = 10(0.00879) = 0.0879

Binomial Probabilities

Larson/Farber Ch. 4

Find the probability of getting exactly 3 questions correct on the quiz.

Each of these 10 ways has a probability of 0.00879.

P(x = 3) = 10(0.25)3(0.75)2= 10(0.00879)= 0.0879

Combination of n values, choosing x

There are ways.

Larson/Farber Ch. 4

Binomial ProbabilitiesIn a binomial experiment, the probability of exactly x successes in n trials is

Use the formula to calculate the probability of getting none correct, exactly one, two, three, four correct or all 5 correct on the quiz.

P(3) = 0.088 P(4) = 0.015 P(5) = 0.001

Larson/Farber Ch. 4

Binomial Distribution

0 1 2 3 4 5

0

.10

.20

.30

.40

.237

.396

.294

.088

.015 .001

x P(x)0 0.2371 0.3962 0.2643 0.0884 0.0155 0.001

Binomial Histogram

x

Larson/Farber Ch. 4

Probabilities

1. What is the probability of answering either 2 or 4 questions correctly?

2. What is the probability of answering at least 3 questions correctly?

3. What is the probability of answering at least one question correctly?

P( x = 2 or x = 4) = 0.264 + 0.015 = 0. 279

P(x 3) = P( x = 3 or x = 4 or x = 5) = 0.088 + 0.015 + 0.001 = 0.104

P(x 1) = 1 - P(x = 0) = 1 - 0.237 = 0.763

x P(x)0 0.2371 0.3962 0.2643 0.0884 0.0155 0.001

Larson/Farber Ch. 4

Parameters for a Binomial Experiment

Use the binomial formulas to find the mean, variance and standard deviation for the distribution of correct answers on the quiz.

Mean:

Variance:

Standard deviation:

Larson/Farber Ch. 4

More Discrete Probability

Distributions

Section 4.3

Larson/Farber Ch. 4

The Geometric DistributionA marketing study has found that the probability that a person who enters a particular store will make a purchase is 0.30.

•The probability the first purchase will be made by the first person who enters the store 0.30. That is P(1) = 0.30.

•The probability the first purchase will be made by the second person who enters the store is (0.70) ( 0.30). So P(2) = (0.70) ( 0.30) = 0.21.

•The probability the first purchase will be made by the third person who enters the store is (0.70)(0.70)( 0.30). So P(3) = (0.70) (0.70) (0.30) = 0.147.

The probability the first purchase will be made by person number x is P(x) = (.70)x - 4(.30)

Larson/Farber Ch. 4

The Geometric Distribution

A geometric distribution is a discrete probability distribution of the random variable x that satisfies the following conditions.

1. A trial is repeated until a success occurs.2. The repeated trials are independent of each

other.3. The probability of success p is the same for

each trial.

The probability that the first success will occur on trial number x is P(x) = (q)x – 1p where q = 1 – p

Larson/Farber Ch. 4

ApplicationA cereal maker places a game piece in its boxes. The probability of winning a prize is one in four. Find the probability youa) Win your first prize on the 4th purchase

b) Win your first prize on your 2nd or 3rd purchase

c) Do not win your first prize in your first 4 purchases.

P(4) = (.75)3 .(.25) = 0.1055

P(2) = (.75)1(.25) = 0.1875 and P(3) = (.75)2(.25) = 0.1406So P(2 or 3 ) = 0.1875 + 0.1406 = 0.3281

1 – (P(1) + P(2) + P(3) + P(4))1 – ( 0.25 + 0.1875 + 0.1406 + 0.1055) = 1 – .6836 = 0.3164

Larson/Farber Ch. 4

The Poisson DistributionThe Poisson distribution is a discrete probability distribution of the random variable x that satisfies the following conditions.1. The experiment consists of counting the number of times, x, an

event occurs in an interval of time, area or space.2. The probability an event will occur is the same for each interval.3. The number of occurrences in one interval is independent of the

number of occurrences in other intervals.

The probability of exactly x occurrences in an interval is

e is the irrational number approximately 2.71828 is the mean number of occurrences per interval.

Larson/Farber Ch. 4

Application

a) Three people are killed by sharks this year

b) Two or three people are killed by sharks this year

P(3) = 0.0076

P(2 or 3) = 0.0023 + 0.0076 = 0.0099

It is estimated that sharks kill 10 people each year worldwide. Find the probability