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4 Discrete Probability Distributions
x = number of on time
arrivals
x = number of
correct answers
Elementary Statistics
Larson
Farber
x = number of employees
reaching sales quota
Larson/Farber Ch. 4
x = number of
points scored in a
game
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 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.
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
3
P(x)
0.004
0.435
0.355
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
Number of Vehicles
0.435
.40
0.355
P(x)
.30
0.206
.20
.10
0.004
0
00
11
22
33
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. 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.
x
0
1
2
3
frequency
2
217
178
103
Larson/Farber Ch. 4
P(x)
0.004
0.435
0.355
0.206
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
P(x)
0.004
0.435
0.355
0.206
x- μ
-1.763
-0.763
0.237
1.237
(x -μ )
3.108
0.582
0.056
1.530
P(x)(xP(x)
- )
0.012
0.253
0.020
0.315
0.601
variance
The standard deviation is 0.775 vehicles.
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
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
Larson/Farber Ch. 4
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 trials:
x = 0, 1, 2, 3, … , n.
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 = 0.80
Larson/Farber Ch. 4
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
FFSSS
SSFSF SSFFS SFFSS SFSFS
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(3) = 0.088
Larson/Farber Ch. 4
P(4) = 0.015
P(x)
0.237
0.396
0.264
0.088
0.015
0.001
P(5) = 0.001
Binomial Distribution
x
0
1
2
3
4
5
Binomial Histogram
.396
.40
.30
.294
.237
P(x)
0.237
0.396
0.264
0.088
0.015
0.001
.20
.088
.10
.015
.001
4
5
0
0
Larson/Farber Ch. 4
1
2
3
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?
3. What is the probability of answering at least one question correctly?
Larson/Farber Ch. 4
P(x)
0.237
0.396
0.264
0.088
0.015
0.001
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
P(x)
0.237
0.396
0.264
0.088
0.015
0.001
2. What is the probability of answering at least 3 questions correctly?
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
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?
Larson/Farber Ch. 4