Larson/Farber Ch. 4

Elementary Statistics

Larson 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

Definitions

• probability distribution– discrete probability distribution (Chapter 4)– continuous probability distribution (Chapter 5 +)

• random variable– discrete random variable– continuous random variable

• mean of a probability distribution

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

Types of Random Variables

X = Number of sales calls a salesperson makes in one day

P(x>10)

x = Hours spent on sales calls in one day.

P(X >6.5)

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

Discrete or Continuous?

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

Constructing a Discrete Probability Distribution

1. Make a frequency distribution for the possible outcomes.

2. Find the sum of the frequencies.

3. Find the probability of each possible outcome by dividing its frequency by the sum of the frequencies.

4. Check that each probability is between 0 and 1 and that the sum is 1.

Larson/Farber Ch. 4

x frequency P(x)0 2 0.0041 217 0.4352 178 0.3553 103 0.206

Discrete Probability Distributions

1. Make a frequency distribution for the possible outcomes.

2. Find the sum of the frequencies.

3. Find the probability of each possible outcome by dividing its frequency by the sum of the frequencies.

4. Check that each probability is between 0 and 1 and that the sum is 1.

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:

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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

Expected Value

Expected value of a discrete random variable

• Equal to the mean of the random variable.• E(x) = μ = ΣxP(x)

Interpretation: We would EXPECT each household in this population to own 1.8 cars (std dev = .775 cars)

Larson/Farber Ch. 4

Binomial Distributions

Section 4.2

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

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

conditions.• Each trial has 2 outcomes• There is a fixed probability of success on a single trial.

Binomial Experiments

4 Characteristics of a Binomial Experiment

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

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 Experiments

Characteristics of a Binomial Experiment

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

Larson/Farber Ch. 4

Notation – Binomial experiments

Symbol Description

n The number of times a trial is repeated

p = P(s) The probability of success in a single trial

q = P(F) The probability of failure in a single trial (q = 1 – p)

x The random variable represents a count of the number of successes in n trials: x = 0, 1, 2, 3, … , n.

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.

Binomial Experiments

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 SSSFF

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

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?

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

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

Can I do this in Excel?

Larson/Farber Ch. 4

Can I do this in Excel?

Larson/Farber Ch. 4

Can I do this in Excel?