4 Discrete Probability Distributions

x

= number of correct answers

x

= number of on time arrivals

Elementary Statistics

Larson Farber

x

= number of employees reaching sales quota

x

= number of points scored in a game 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

Random Variables

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 Larson/Farber Ch. 4

Types of Random Variables

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 value within an interval. The possible outcomes cannot be listed. Continuous random variables are determined by a measure.

continuous if it can take on any 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

Discrete or Continuous?

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 Larson/Farber Ch. 4

Types of Random Variables

Identify each random variable as discrete or continuous.

x

= The number of people in a car Discrete –

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

x

= The gallons of gas bought in a week Continuous –

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

x

= The time it takes to drive from home to school Continuous –

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

x

= The number of trips to school you make per week Discrete –

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

Larson/Farber Ch. 4

Discrete Probability Distributions

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.

number of vehicles

x

0 1 2 P(x) 0.004

0.435

0.355

3 0.206

Properties of a probability distribution

Each probability must be between 0 and 1, inclusive.

• The sum of all probabilities is 1.

Larson/Farber Ch. 4

Probability Histogram

.40

.30

.20

Number of Vehicles 0.435

0.355

0.206

.10

0.004

0 0 0 1 1 2 2 3 3 x • 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.

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

Discrete Probability Distributions

1.

2.

3.

4.

Make a frequency distribution for the possible outcomes.

Find the sum of the frequencies.

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

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

x

0 1 2 3

frequency

2 217 178 103 P(x) 0.004

0.435

0.355

0.206

Larson/Farber Ch. 4

Mean, Variance and Standard Deviation

The mean of a discrete probability distribution is: The variance of a discrete probability distribution is: The standard deviation of a discrete probability distribution is: Larson/Farber Ch. 4

Mean (Expected Value)

Calculate the mean Multiply each value by its probability. Add the products x 0 1 2 3 P(x) 0.004

0.435

0.355

0.206

xP(x) 0 0.435

0.71

0.618

1.763

The expected value (the mean) is 1.763 vehicles.

Larson/Farber Ch. 4

Calculate the Variance and Standard Deviation

The mean is 1.763 vehicles.

x 0 1 2 3 Larson/Farber Ch. 4 P(x) 0.004

0.435

0.355

0.206

x μ -1.763

-0.763

0.237

1.237

3.108

0.582

0.056

1.530

0.012

0.253

0.020

0.315

0.601

variance The standard deviation is 0.775 vehicles.

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

Section 4.2

Binomial Distributions

Larson/Farber Ch. 4

Guess the Answers

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 Larson/Farber Ch. 4

Quiz Results

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

?

Larson/Farber Ch. 4

Binomial Experiments

4 Characteristics of a Binomial Experiment • 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. The random variable

x

is a count of the number of successes in

n

trials.

Larson/Farber Ch. 4

Binomial Experiments

Characteristics of a Binomial Experiment • 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

. The random variable

x

is a count of the number of successes in

n

trials.

Larson/Farber Ch. 4

Notation – Binomial experiments

Symbol n p = P(s) q = P(F) x Description

The number of times a trial is repeated The probability of success in a single trial ( The probability of failure in a single trial

q

= 1 –

p

) The random variable represents a count of the number of successes in

n x

= 0, 1, 2, 3, … ,

n

.

trials: Larson/Farber Ch. 4

Binomial Experiments

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

.

Larson/Farber Ch. 4

Binomial Experiments

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

.

n

= 8

p

= 1/3

q

= 2/3

x

= 5 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

= 7

p

Larson/Farber Ch. 4 = 0.80

q

= 0.20

x

= 6

Binomial Probabilities

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 SFSFS FFSSS 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

Larson/Farber Ch. 4

Combination of n values, choosing x

There are ways.

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

Larson/Farber Ch. 4

Binomial Probabilities

In 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 Larson/Farber Ch. 4 P(4) = 0.015 P(5) = 0.001

Binomial Distribution

x 0 1 2 3 4 5 P(x) 0.237

0.396

0.264

0.088

0.015

0.001

P(3) = 0.088 Larson/Farber Ch. 4 P(4) = 0.015 P(5) = 0.001

Binomial Histogram .40

.396

.30

.237

Binomial Distribution

.294

x 0 1 2 3 4 5 P(x) 0.237

0.396

0.264

0.088

0.015

0.001

.20

.088

.10

.015

.001

0 0 1 2 3 4 5 Larson/Farber Ch. 4

x

Probabilities

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

x 0 1 2 3 4 5 2. What is the probability of answering at least 3 questions correctly?

P(x) 0.237

0.396

0.264

0.088

0.015

0.001

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

Larson/Farber Ch. 4

Probabilities

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

P(

x

= 2 or

x

= 4) = 0.264 + 0.015 = 0. 279 x 0 1 2 3 4 5 2. What is the probability of answering at least 3 questions correctly?

P(x) 0.237

0.396

0.264

0.088

0.015

0.001

P(

x

 3) = P(

x

= 3 or

x

= 4 or

x

= 5) = 0.088 + 0.015 + 0.001 = 0.104 3. What is the probability of answering at least one question correctly?

P(

x

 1) = 1 - P(

x

= 0) = 1 - 0.237 = 0.763

Larson/Farber Ch. 4

Parameters for a Binomial Experiment

Mean: Variance: Standard deviation:

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

Larson/Farber Ch. 4

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?