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Understandable Statistics
Seventh Edition
By Brase and Brase
Prepared by: Lynn Smith
Gloucester County College
Chapter Five
The Binomial Probability Distribution
and Related Topics
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1
Statistical Experiment
A statistical experiment or
observation is any process by which
an measurements are obtained
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2
Examples of Statistical
Experiments
• Counting the number of books in the
College Library
• Counting the number of mistakes on a
page of text
• Measuring the amount of rainfall in your
state during the month of June
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3
Random Variable
a quantitative variable that
assumes a value determined by
chance
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4
Discrete Random Variable
A discrete random variable is a
quantitative random variable that can
take on only a finite number of values or
a countable number of values.
Example: the number of books in the
College Library
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5
Continuous Random Variable
A continuous random variable is a
quantitative random variable that can
take on any of the countless number of
values in a line interval.
Example: the amount of rainfall in your
state during the month of June
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6
Probability Distribution
an assignment of probabilities to
the specific values of the random
variable or to a range of values of
the random variable
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7
Probability Distribution of a
Discrete Random Variable
• A probability is assigned to each value of
the random variable.
• The sum of these probabilities must be 1.
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8
Probability distribution for the
rolling of an ordinary die
x
1
2
3
P(x)
1
6
1
6
1
6
4
1
6
5
1
6
6
1
6
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9
Features of a Probability
Distribution
x
P(x)
1
1
6
2
1
6
3
1
6
4
1
6
5
1
6
6
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Probabilities
must be between
zero and one
(inclusive)
1
6
6
 1
6
 P(x) =1
10
Probability Histogram
P(x)
1
6
|
|
1
|
2
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|
3
|
4
|
5
|
6
11
Mean and standard deviation
of a discrete probability
distribution
Mean =  = expectation or expected value,
the long-run average
Formula:
 =  x P(x)
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12
Standard Deviation

2
(
x


)
 P( x )

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13
Finding the mean:
x
P(x)
0
.3
0
1
.3
.3
2
.2
.4
3
.1
.3
4
.1
.4
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x P(x)
1.4
 =  x P(x) =
1.4
14
Finding the standard deviation
x –
( x – ) 2
( x – ) 2 P(x)
x
P(x)
0
.3
– 1.4
1.96
.588
1
.3
– 0.4
0.16
.048
2
.2
.6
0.36
.072
3
.1
1.6
2.56
.256
4
.1
2.6
6.76
.676
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1.64
15
Standard Deviation

 ( x   )  P ( x )  1.64 
2
1.28
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16
Linear Functions of a Random
Variable
If a and b are any constants and x is a random
variable, then the new random variable
L = a + bx
is called a linear function of a random
variable.
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17
If x is a random variable with
mean  and standard deviation
, and L = a + bx then:
• Mean of L =  L = a + b 
• Variance of L = L 2 = b2  2
• Standard deviation of L =  L= the square
root of b2  2 = b 
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18
If x is a random variable with
mean = 12 and standard
deviation = 3 and L = 2 + 5x
• Find the mean of L.
• Find the variance of
L.
• Find the standard
deviation of L.
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• L = 2 + 5

• Variance of L = b2  2 =
25(9) = 225
• Standard deviation of L
= square root of 225 =

19
Independent Random Variables
Two random variables x1 and x2 are
independent if any event involving
x1 by itself is independent of any
event involving x2 by itself.
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20
If x1 and x2 are a random variables
with means  and and variances
 and  If W = ax1 + bx2 then:
• Mean of W =  W = a  + b 
• Variance of W = W 2 = a2  12 + b2  2
• Standard deviation of W =  W= the
square root of a2  12 + b2  2
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21
Given x1, a random variable with
1 = 12 and  1 = 3 and x2 is a random
variable with  2 = 8 and  2 = 2 and
W = 2x1 + 5x2.
• Find the mean of W.
• Find the variance of
W.
• Find the standard
deviation of W.
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• Mean of W =
2(12) + 5(8) = 64
• Variance of W =
4(9) + 25(4) = 136
• Standard deviation
of W= square root of
136  11.66
22
Binomial Probability
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23
Features of a Binomial
Experiment
1. There are a fixed number of trials.
We denote this number by the letter n.
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24
Features of a Binomial
Experiment
2.
The n trials are independent and
repeated under identical conditions.
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25
Features of a Binomial
Experiment
3.
Each trial has only two outcomes:
success, denoted by S,
and failure, denoted by F.
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26
Features of a Binomial
Experiment
4. For each individual trial, the
probability of success is the same.
We denote the probability of success by p
and the probability of failure by q.
Since each trial results in either success or
failure, p + q = 1 and q = 1 – p.
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27
Features of a Binomial
Experiment
5. The central problem is to find the
probability of r successes out of n trials.
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28
Binomial Experiments
• Repeated, independent trials
• Number of trials = n
• Two outcomes per trial: success (S)
and failure (F)
• Number of successes = r
• Probability of success = p
• Probability of failure = q = 1 – p
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29
A sharpshooter takes eight
shots at a target. She normally
hits the target 70% of the time.
Find the probability that she
hits the target exactly six times.
Is this a binomial experiment?
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30
Is this a binomial experiment?
A sharpshooter takes eight shots at a
target. She normally hits the target 70%
of the time. Find the probability that she
hits the target exactly six times.
success =
failure =
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31
Is this a binomial experiment?
A sharpshooter takes eight shots at a
target. She normally hits the target 70%
of the time. Find the probability that she
hits the target exactly six times.
success = hitting the target
failure = not hitting the target
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32
Is this a binomial experiment?
A sharpshooter takes eight shots at a
target. She normally hits the target 70%
of the time. Find the probability that she
hits the target exactly six times.
Probability of success =
Probability of failure =
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33
Is this a binomial experiment?
A sharpshooter takes eight shots at a
target. She normally hits the target 70%
of the time. Find the probability that she
hits the target exactly six times.
Probability of success = 0.70
Probability of failure = 1 – 0.70 = 0.30
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34
Is this a binomial experiment?
A sharpshooter takes eight shots at a
target. She normally hits the target 70%
of the time. Find the probability that she
hits the target exactly six times.
In this experiment there are n = _____
trials.
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35
Is this a binomial experiment?
A sharpshooter takes eight shots at a
target. She normally hits the target 70%
of the time. Find the probability that she
hits the target exactly six times.
In this experiment there are n = _8__ trials.
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36
Is this a binomial experiment?
A sharpshooter takes eight shots at a
target. She normally hits the target 70%
of the time. Find the probability that she
hits the target exactly six times.
We wish to compute the probability of six
successes out of eight trials. In this case
r = _____.
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37
Is this a binomial experiment?
A sharpshooter takes eight shots at a
target. She normally hits the target 70%
of the time. Find the probability that she
hits the target exactly six times.
We wish to compute the probability of six
successes out of eight trials. In this case
r = _ 6__.
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38
Binomial Probability Formula
P (r )  C n , r p q
r
where Cn , r  binomial
n r
coefficient
n!

r!(n  r )!
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39
Calculating Binomial
Probability
Given n = 6, p = 0.1, find P(4):
6!
4
2
P( 4) 
(.1) (.9) 
4 !( 6  4) !
15(.0001)(.81)  0.001215
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40
Calculating Binomial
Probability
A sharpshooter takes eight shots at a target. She
normally hits the target 70% of the time. Find the
probability that she hits the target exactly six times.
n = 8, p = 0.7, find P(6):
8!
6
2
P( 6) 
(.7 ) (.3) 
6!2!
28(.117649)(.09)  0.2964755
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41
Table for Binomial Probability
Table 3
Appendix II
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42
Using the Binomial Probability
Table
• Find the section labeled with your value
of n.
• Find the entry in the column headed with
your value of p and row labeled with the r
value of interest.
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43
Using the Binomial Probability
Table
n = 8, p = 0.7, find P(6):
p
.70
n
8
r
.
4
5
.254
6
.296
7
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.198
44
Find the Binomial Probability
Suppose that the probability that a certain
treatment cures a patient is 0.30. Twelve
randomly selected patients are given the
treatment. Find the probability that:
a.
b.
c.
d.
exactly 4 are cured.
all twelve are cured.
none are cured.
at least six are cured.
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45
Exactly four are cured:
n=
r=
p=
q=
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46
Exactly four are cured:
n = 12
r= 4
P(4) = 0.231
p = 0.3
q = 0.7
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47
All are cured:
n = 12
r = 12
p = 0.3
P(12) = 0.000
q = 0.7
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48
None are cured:
n = 12
r= 0
p = 0.3
P(0) = 0.014
q = 0.7
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49
At least six are cured:
r= ?
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50
At least six are cured:
r = 6, 7, 8, 9, 10, 11, or 12
P(6) = .079
P(10) = .000
P(7) = .029
P(11) = .000
P(8) = .008
P(12) = .000
P(9) = .001
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51
At least six are cured:
P( 6, 7, 8, 9, 10, 11, or 12)
= .079 + .029 + .008 + .001 + .000 +
.000 + .000
= .117
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52
Graph of a Binomial
Distribution
Binomial distribution
for n = 4, p = 0.4:
r
P( r)
0
.1296
1
.3456
2
.3456
3
.1536
4
.0256
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53
Graph of a Binomial
Distribution
Binomial distribution
for n = 4, p = 0.4:
P( r )
r
P( r)
.4
0
.1296
.3
1
.3456
2
.3456
3
.1536
4
.0256
.2
.1
0
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1
2
3
4
r
54
Mean and Standard Deviation
of a Binomial Distribution
Mean    np
Standard Deviation    npq
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55
Mean and standard deviation
of the binomial distribution
Find the mean and standard deviation of
the probability distribution for tossing four
coins and observing the number of heads
appearing.
n=4
p = 0.5
q = 1 – p = 0.5
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56
Mean and standard deviation
of the binomial distribution
Mean    np  4(.5)  2
Standard Deviation    npq 
4(.5)(.5)  1  1
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57
Find the mean and standard
deviation:
A sharpshooter takes eight shots at a target. She
normally hits the target 70% of the time. Find the
probability that she hits the target exactly six times.
n = 8, p = 0.7
  np  8(.7)  5.6
  npq  8(.7)(.3)  1.68  1.3
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58
Geometric Distribution
A probability distribution to
determine the probability that
success will occur on the nth trial of
a binomial experiement
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59
Geometric Distribution
• Repeated binomial trials
• Continue until first success
• Find probability that first success comes
on nth trial
• Probability of success on each trial = p
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60
Geometric Probability
Probabilit y of success on nth trial 
P ( n)  p(1  p ) n1
1
mean   
p
1-p
standard deviation   
p
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61
A sharpshooter normally hits
the target 70% of the time.
• Find the probability that her first hit is
on the second shot.
• Find the mean and the standard
deviation of this geometric distribution.
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62
A sharpshooter normally hits
the target 70% of the time.
• Find the probability that her first hit is
on the second shot.
P(2)=p(1-p) n-1 = .7(.3)2-1 = 0.21
• Find the mean
 = 1/p = 1/.7 1.43
• Find the standard deviation
 
1-p

p
1  .7
 0 . 78
.7
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63
Poisson Distribution
A probability distribution where
the number of trials gets larger and
larger while the probability of
success gets smaller and smaller
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64
Poisson Distribution
• Two outcomes : success and failure
• Outcomes must be independent
• Compute probability of r occurrences in
a given time, space, volume or other
interval
  (Greek letter lambda) represents mean
number of successes over time, space,
area
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65
Poisson Distribution
e 
Probabilit y of r successes  P ( r ) 
r!
where e  2.7183 and
  mean number of successes over time, volume, area, etc.
Population mean    

r
Population standard deviation    
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66
The mean number of people
arriving per hour at a shopping
center is 18.
• Find the probability that the number of
customers arriving in an hour is 20.
r = 20
 = 18
e = 2.7183
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Find P(20)
67
The mean number of people
arriving per hour at a shopping
center is 18.
18 e
P ( 20) 
 0.0798
20!
There is almost an 8% chance that
twenty people will arrive in an
hour.
20
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18
68
Poisson Probability
Distribution Table
Table 4 in Appendix II provides the
probability of a specified value of r
for selected values of .
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69
Using the Poisson Table
 = 18, find P(20):

r
…
18
19
20
21
..
.
.
.
17
…
.0909
.0814
.0692
18
19
…
.0936
.0887
.0798
.0560 .0684
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70
Poisson Approximation to the
Binomial Distribution
The Poisson distribution can be
used as a probability distribution
for “rare” events.
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71
“Rare” Event
The number of trials (n) is large
and the probability of success (p) is
small.
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72
If n  100 and np < 10, then
• The distribution of r (the number of
successes) has a binomial distribution
which is approximated by a Poisson
distribution .
• The mean  = np.
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73
Use the Poisson distribution to
approximate the binomial
distribution:
• n = 240
• p = 0.02
• Find the probability of at most 3
successes.
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74
Using the Poisson to approximate
the binomial distribution for n =
240 and p = 0.02
Note that n  100 and np = 4.8 < 10, so the
Poisson distribution can be used to
approximate the binomial distribution.
Find the probability of at most 3 successes:
Since  = np = 4.8, we use Table 4 to find
P( r  3) =.0082 + .0395 + . 0948 + . 1517 = .2942
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75