Discrete Probability Distributions

Elementary Statistics

Larson Farber

x = number of on time arrivals

x = number of points scored in a game

x = number of employees reaching quota

x = number of correct answers

Chapter4

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

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.

Random Variables

Two types of random variables

Types of Random Variables

• 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.

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.

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. ( ) 1P x 0 ( ) 1P x

Probability Histogram

• 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

Mean, Variance and Standard Deviation

The variance of a discrete probability distribution is:

)()( 22 xPx

The standard deviation of a discrete probability distribution is:

2

The mean of a discrete probability distribution is:

)(xPx

Mean (Expected value)

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

1.763

Multiply each value times its probability. Add the products

)(xPx

The expected value (the mean) is 1.763 vehicles.

Calculate the mean

Variance and Standard Deviation

2 0.601 0.775 The standard deviation is 0.775 vehicles.

)()( 22 xPx The mean is 1.763 vehicles.

Calculate the varianceCalculate the standard deviation

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 μ 2

variance

• 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 and the probability of failure is q. P(S) = p P(F) =q 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

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

1. What is the 11th digit after the decimal point for the irrational number e?(a) 2 (b) 7 (c) 4 (d) 52. 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) 22404. 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

Quiz

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

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

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

Denote 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 SFFSS

There are 1012123

12345

!2!3

!5

)!(!

!

xnx

nCxn

ways.

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 Probability

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

xnxxnxxn qp

xnx

nqpCxP

)!(!

!)(

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

237.0)75(.)25(.)!05(!0

!5)0( 50050

05

qpCP

396.0)75(.)25(.)!15(!1

!5)1( 41151

15

qpCP

264.0)75(.)25(.)!25(!2

!5)2( 32252

25

qpCP

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

Binomial Distribution

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

Binomial Histogram

x

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

Calculate the mean, variance and standard deviation

x P(x) xP(x)0 0.237 0.0001 0.396 0.3962 0.264 0.5273 0.088 0.2644 0.015 0.0595 0.001 0.005

1.25

2( ) ( ) 0.9375x P x ( ) 1.25xP x

0.9375 0.968 The expected value (mean) is 1.25, the variance is 0.9375, and the standard deviation Is 0.968

-1.25 1.5625 0.371-0.25 0.0625 0.0250.75 0.5625 0.1481.75 3.0625 0.2692.75 7.5625 0.1113.75 14.0625 0.014

x 2( )x 2( ) ( )x P x

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.

968.09375.2 npq

2 5(.25)(.75) 0.9375npq 5(.25) 1.25np

Mean:

Variance :

Standard deviation:

np

npq

2 npq