Geometric Distributions
AP Statistics – Section 8.2
In some ways, the geometric distribution resembles the binomial distribution. It deals with a series of independent trials,
each of which has two possible outcomes, success or failure. The difference between the two distributions lies in the
definition of the underlying random variable, X. In a binomial distribution, X represents the number of successes in n trials,
whereas in a geometric distribution, X represents the number of the trial on which the first success occurs.
I. Geometric Distribution (“waiting time”) Formula
Suppose there is a series of independent trials of an experiment, where p is the probability of success on one of these trials.
Let the random variable X represent the number of the trial on which the first success occurs. The probability that the first
success is obtained on the xth trial is given by the formula
P(X = x) = (1 - p)x-1p
Example 1 (Statistics in Action):
Suppose 9% of the engines manufactured on a certain assembly line have at least one defect. Engines are randomly sampled
from this line one at a time and tested. What is the probability that the first nondefective engine is found
(a) on the third trial
(b) before the fourth trial
(c) after the fifth trial
Example 2 (Statistics in Action):
About 10% of the U.S. population has type B blood. Suppose our medical technician is checking donations that may be
considered independent with respect to blood type.
(a) What is the probability that the first donation that is type B is the third one checked?
(b) What is the probability that he will have to check at most three donations to get the first that is type B?
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(c) What is the probability that he will have to check at least three donations to get the first that is type B?
(d) Part (c) refers to the memoryless property of the geometric distribution. Explain why that term is appropriate.
Example 3
In a geometric distribution, what are the possible values of x? Give your answer as precisely as possible.
Example 4
Use your graphing calculator to graph the geometric distribution for p=.4 and sensible values of x, under the assumption that
the distribution is continuous (which, of course, it is not). What should the graph really look like?
Example 5
Why is this distribution called “geometric”?
II. Mean and Standard Deviation of a Geometric Distribution
(1) The mean (expected value) of a geometric distribution is
1
=p
(this represents the expected number (average) number of trials that would occur before the first success appears )
(2) The standard deviation of a geometric distribution is
1-p
=
p2
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Example 6 (Statistics in Action):
The probability that a random blood donation is type B is 0.1
(a) What is the expected number of donations that have to be checked to obtain the first one that is type B?
(b) What is the standard deviation of the number of donations checked to obtain the first one of type B?
(c) What is the expected number of donations checked to obtain two that are type B?
Example 7 (Statistics in Action):
The phone line for a 24-hour ticket office is busy 60% of the time.
(a) If you dial at random times throughout the week, what is the expected number of tries it will take you to get
through? What is the standard deviation of the number of tries?
(b) What is the expected total number of times you will have to dial if you forgot to ask a question after getting through
the first time and have to call again?
Homework: #8.37-8.39, 8.41, 8.43-8.45, 8.47
Chapter 8 Review Problems: #8.55-8.57, 8.59-8.63
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