Chapter 4
Probability Distributions
4-1 Random Variables
4-2 Binomial Probability Distributions
4-3 Mean, Variance, Standard Deviation
for the Binomial Distribution
4-4 Other Discrete Probability Distributions
1
Overview
This chapter will deal with the
construction of
probability distributions
by combining the methods of Chapter 2
with the those of Chapter 3.
Probability Distributions will describe
what will probably happen instead of
what actually did happen.
2
Combining Descriptive Statistics Methods and
Probabilities to Form a Theoretical Model of
Behavior
3
4-1
Random Variables
4
Definitions
 Random Variable
a variable (typically represented by x) that has a
single numerical value, determined by chance,
for each outcome of a procedure
Probability Distribution
a graph, table, or formula that gives the
probability for each value of the random variable
5
Probability Distribution
Number of Girls Among Fourteen Newborn Babies
x
P(x)
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
0.000
0.001
0.006
0.022
0.061
0.122
0.183
0.209
0.183
0.122
0.061
0.022
0.006
0.001
0.000
Questions:
Let X = # of girls
1.
P(X=3) = ?
2.
P(X ≤ 2) = ?
3.
P(X ≥ 1) = ?
6
Probability Histogram
7
Definitions
Discrete random variable
has either a finite number of values or countable
number of values, where ‘countable’ refers to the
fact that there might be infinitely many values,
but they result from a counting process.
Continuous random variable
has infinitely many values, and those values can
be associated with measurements on a
continuous scale with no gaps or interruptions.
8
Requirements for
Probability Distribution
 P(x) = 1
where x assumes all possible values
0  P(x)  1
for every value of x
See #3 on hw
9
Mean, Variance and Standard Deviation
of a Probability Distribution
Formula (Mean)
µ =  [x • P(x)]
Formula (Variance)
2
2
 =  [(x - µ) • P(x)]
Formula (Standard Deviation)
2
 = [(x - µ) • P(x)]
10
Roundoff Rule for µ,  , and 
2
Round results by carrying one more decimal
place than the number of decimal places used
for the random variable x. If the values of x
are integers, round µ, 2, and  to one
decimal place.
11
TI-83 Calculator
Calculate Mean and Std. Dev from a Probability
Distribution
1. Press Stat
2. Press “1” Edit
3. Enter values of random variable (x) in L1
4. Enter probability P(x) in L2
5. Press Stat
6. Cursor over to CALC
7. Choose the 1-Var stats option
8. Enter 1-Var stats L1,L2
12
Using Excel
See Probability Distribution Worksheet
Examples:
1. See Introduction to probability distributions handout
2. Go to Excel (dice example – class assignment)
3. Find missing probability, mean, SD and unusual values
(test question see #6 on hw)
13
Unusual and Unlikely Values
1. Unusual if greater than 2 standard deviations
from the mean, that is x + 2 and x – 2
2. Unlikely if probability is very small, usually
less than .05. This is consistent with the 2
idea associated with the empirical rule.
#6f and 7c on HW
14
Definition
Expected Value
The average value of outcomes
E =  [x • P(x)]
15
E =  [x • P(x)]
Example: #9 on hw
x • P(x)
Event
x
P(x)
Win
$5
244/495
2.465
Lose
- $5
251/495
- 2.535
E = -$.07
16
4-2
Binomial Random
Variables
17
Binomial Random Variables
Facts:
Discrete (we can count the outcomes)
Have to do with random variables having 2
outcomes. Examples: heads/tails, boy/girl, yes/no,
defective/not defective, etc.
A binomial distribution is the sum of several trials.
Example: # of heads when a coin is tossed three times
18
Notation for Binomial Probability
Distributions
n =
fixed number of trials
x = specific number of successes in n trials
p = probability of success in one of n trials
q = probability of failure in one of n trials
(q = 1 - p )
P(x) = probability of getting exactly x successes among n
trials
19
Method 1
Binomial Probability
Formula
 P(x) =
n!
•
(n - x )! x!
 P(x) = nCx • px
px •
•
n-x
q
qn-x
for calculators use nCr key, where r = x
20
Binomial Probability
Formula
P(x) =
n!
•
(n - x )! x!
Number of
outcomes with
exactly x
successes
among n trials
px •
n-x
q
Probability of x
successes
among n trials
for any one
particular order
21
Example: Toss a coin 3 times.
Let x = number of heads and find
a) P(2) =
b) P(at least 2)
This is a binomial experiment so you need to know 4
things p, q, n and x.
p=.5
q=.5
n=3
a) x = 2
b) x = 0 then 1 then 2
On the test you will have to construct the entire probability
distribution for tossing a coin “n” times and observing the number of
heads. Should do this before the test.
22
Example: Find the probability of getting exactly
2 correct responses among 5 different requests
from directory assistance. Assume in general,
they are correct 80% of the time.
This is a binomial experiment where:
n=5
x=2
p = 0.80
q = 0.20
Using the binomial probability formula to solve:
P(2) =
5C2
2
3
• 0.8 • 0.2 = 0.0512
23
Method 2
Binomial Table
Two tables are available on website
24
Example: Using Table for n = 5 and p = 0.80,
find the following:
a) The probability of exactly 2 successes
b) The probability of at most 2 successes
c) The probability of at least 1 success
a) P(2) = 0.0512
Test Question
b) P(at most 2) = P(0) or P(1) or P(2)
= 0.0003 + 0.0064 + 0.0512
= 0.0579
c) P(at least 1) = 1 – P(0) = 1 – .0003 = .9997
25
Method 3
Using Technology
Calculator function (TI-83)
See binomial distribution worksheet
See also coin example worksheet
26
TI-83 Calculator
Finding Binomial Probabilities (complete distribution)
1. Press 2nd Distr
2. Choose binopdf
3. Enter binopdf(n,p)
4. Press STO L2 (stores probabilities in column L2)
5. Press Stat
6. Choose Edit (to view probabilities)
7. Optional: enter the values of the random variable in
L1
27
TI-83 Calculator
Finding Binomial Probabilities (individual value)
1. Press 2nd Distr
2. Choose binopdf
3. Enter binopdf(n,p,x)
28
TI-83 Calculator
Finding Binomial Probabilities (cummulative)
1. Press 2nd Distr
2. Choose binocdf
3. Enter binocdf(n,p,x)
This yields the sum of the probabilities from 0 to x.
Example:
Let n=6 and p=0.2
P(X<3) = binocdf(6,.2,3)
29
4.3 Mean, variance and
standard deviation of a
Binomial Probability
Distribution
30
For Any Discrete Probability
Distribution the general formulas
are:
• µ = [x • P(x)]
• 
2
=  [(x - µ) • P(x) ]
2
 =
 [(x - µ) • P(x)]
2
31
For Binomial Distributions:
• µ
=n•p
•  = n • p • q
2
=
n•p•q
32
Example:
Find the mean and standard
deviation for students that guess answers on a
multiple choice test with 5 answers and 20
questions.
• We previously discovered that this scenario could be
considered a binomial experiment where:
• n = 20
• p = 0.2
• q = 0.8
• Using the binomial distribution formulas:
µ = (20)(0.2) = 4 correct answers
=
(20)(0.2)(0.8) = 1.8 answers (rounded)
Test question
33
Reminder
• Maximum usual values = µ + 2 
• Minimum usual values = µ - 2 
34
Example:
Determine whether guessing 7
correct answers is unusual.
For this binomial distribution,
• µ = 4 answers
•
•
•
= 1.8 answers
µ + 2  = 4 + 2(1.8) = 7.6
µ - 2  = 4 - 2(1.8) = .4
The usual number of correct answers would be from .4 to 7.6,
so guessing 7 correct answers would not be an unusual result.
Test question
35
4.4 Other Discrete
Probability Distributions
• Poisson
• Geometric
• Hypergeometric
• Negative Binomial
• And more
36
Poisson Distribution
Definition
a discrete probability distribution that
applies to occurrences of some event
over a specific interval.
Will be a question on the test for you to differentiate
between a binomial and a poisson distribution
37
Definition
Poisson Distribution
a discrete probability distribution that
applies to occurrences of some event
over a specific interval.
Probability Formula
P(x) =
µ x • e -µ where e  2.71828
x!
38
Example:
Look at #1
• Why is this a poisson distribution?
• µ=5
• We need to find various probabilities using
x
-µ
µ
•
e
P(x) =
x!
• Let’s find P(7)
• Look at the Poisson function in Excel
39
Geometric Distribution
Definition
a discrete probability distribution of
the number of trials needed to get one
success.
Will be a question on the test for you to differentiate
between a binomial, poisson and a geometric distribution
40
Geometric Distribution
Example:
Roll a die 5 times. What is the
probability of getting your first 2 on the
5th roll.
41
Negative Binomial
Distribution
Definition
a discrete probability distribution of
the number of trials needed to get a
get a specified number of successes.
42
Negative Binomial
Distribution
Example
a basketball player has a 70% chance
of making a free throw, what is the
probability of making his 3rd free throw
on his 5th shot.
43
Hypergeometric
Distribution
Hypergeometric Experiment
• A sample of size n is randomly selected without
replacement from a population of N items.
. In the population, k items can be classified as
successes, and N - k items can be classified as
failures.
44
Hypergeometric
Distribution
Notation
• N: The number of items in the population.
• k: The number of items in the population that are classified
as successes.
• n: The number of items in the sample.
• x: The number of items in the sample that are classified as
successes.
• kCx: The number of combinations of k things, taken x at a
time.
45
Hypergeometric
Distribution
Example
Suppose we randomly select 5 cards without
replacement from an ordinary deck of playing
cards. What is the probability of getting
exactly 2 red cards (i.e., hearts or diamonds)?
P = [ kCx ] [ N-kCn-x ] / [ NCn ]
= [ 26C2 ] [ 26C3 ] / [ 52C5 ]
= [ 325 ] [ 2600 ] / [ 2,598,960 ] = 0.32513
46