Transcript 8.2 The Geometric Distribution

Daniel S. Yates
The Practice of Statistics
Third Edition
Chapter 8:
The Binomial and
Geometric Distributions
8.2 The Geometric Distribution
Copyright © 2008 by W. H. Freeman & Company
8.2 The Geometric Distribution
1.
2.
3.
4.
What is the geometric setting?
How do you calculate the probability of
getting the first success on the nth trial?
How do you calculate the means and
variance of a geometric distribution?
How do you calculate the probability that
it takes more than n trials to see the first
success for a geometric random
variable?
The Geometric Distribution
•Suppose an experiment consists of a sequence of trials
with the following conditions:
1. The trials are independent.
2. Each trial can result in one of two possible outcomes,
success and failure.
3. The probability of success is the same for all trials.
A geometric random variable is defined as
x = number of trials until the first success is observed
(including the success trial)
The probability distribution of x is called the geometric
probability distribution.
The Geometric Distribution
• If x is a geometric random variable with
probability of success =  for each trial,
then
p(x = n) = (1 – )n-1 
x = 1, 2, 3, …
Example
• Over a very long period of time, it has
been noted that on Friday’s 25% of the
customers at the drive-in window at the
bank make deposits.
• What is the probability that it takes 4
customers at the drive-in window before
the first one makes a deposit.
Example - solution
• This problem is a geometric distribution
problem with  = 0.25.
• Let x = number of customers at the drivein window before a customer makes a
deposit.
The desired probability is
0.1054
p(4)  (.75)41(.25)  0.0117
Geometric Probability
Probabilit y of success on nth trial 
P(n)  p (1  p ) n 1
mean   
1
p
Variance   2 
standard deviation   
1 
2
1- p
p
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.
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
Example
The State Department is trying to identify an
individual who speaks Farsi to fill a foreign embassy
position. They have determined that 4% of the
applicant pool are fluent in Farsi.
a). If applicants are contacted randomly, how many
individuals can they expect to interview in order to find
one who is fluent in Farsi?
1
1
 
 25
 .04
b). What is the probability that they will have to
interview more than 25 until they find one who speaks
Farsi?
n
25
P( X  25)  1     1  0.04  0.3604