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Chapter 8
The Binomial and Geometric
Distributions
8.1 The Binomial Distribution
Big Ideas
1. In the __________________, the random phenomenon is a series of _____
independent trials, each having the same ______________________ of producing
a success. “Success” is the generic term for whatever outcome is of interest to us.
2. The count of ___________________ in the binomial setting has the ___________
distribution with parameters ______ (the ________________________________)
and ______ (the probability of a success on one trial).
3. Probabilities for binomial distributions are best found using the ______________.
4. The _________ of a binomial distribution is ________________ and the standard
deviation is __________________. When the number of trials ____ is
____________, we can approximate binomial probabilities using the Normal
_________________ with this mean and standard deviation.
The Binomial Setting
 n = fixed number of observations
 n observations are _________________
 two outcomes:
 probability of success on any outcome =
Then X = the number of successes in a sample size n has the _______________________
with parameters n and p. Denoted by
Think about it:
What are the possible values for X? Is X a discrete or continuous random variable?
We will learn how to recognize the binomial setting, study the binomial distribution and
how to find binomial probabilities.
Example
Is this a binomial setting?
(a) Observe the gender of the next 50 children born at a local hospital.
X = number of girls
(b) Select one car from each hour’s production for inspection
X = the number of finish defects in paint
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Note about the sample probability p: if we sample with replacement, then removing say
a success item on the first selection will truly change the probability p of a success for the
next selected item – but it is negligible if the population is large relative to the sample.
Example 8.3 Page 515
Note:
Rule of Thumb: population at least 10 times as large as the sample  ok!
Accuracy of this approximation ________________ as the size of the population
___________________ relative to the size of the _________________.
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The Binomial Formula
We will develop the formula together using our probability knowledge.
Company advertises: ships 90% of its orders within three working days. Take a random
sample of 4 orders from that past week. If the company’s claim is true, what is the
probability that …
… all 4 orders are on time?
… all 4 orders are late (0 are on time)?
… 3 orders are on time?
This is the binomial distribution:
P(X = k) =
for k = 0, 1, 2, …,n
where
Try it!
Example Let’s say a company has n = 10 (employed women) and p = 0.25 (proportion
n
never married). Think of it   as: suppose you had n friends, how many ways could
k 
you invite k to dinner?
The ones on the “ends” are easy to do without the formula …
10 
1.   =
0 
3.
2.
4.
10 
  =
9 
3
5. What is the probability of getting exactly 1 success out of 10 independent trials?
6. What is the probability of getting exactly 2 success out of 10 independent trials?
7. What is the probability of getting at least 1 success out of 10 independent trials?
Example 8.7 & 8.8 Page 520
Note: binompdf (n, p, x)
Example 8.10 Page 522
Note: binomcdf (n, p, x)
Aside: To make sure that you get the correct value of X, let’s take a look at an example.
If B(23, 0.7) and we want to know P(X < 17) or P(X  17), then
P(X < 17) =
P(X  17) =
Try question 6 from above: n = 10, p = 0.25, k = 2 probability =
Could actually give the full probability distribution for n = 10, p = 0.25
X
0
1
2
3
4
5
6
7
8
Prob .0563 .1877 .2816
9
10
Could graph it too!
Q: In a random sample of 10 women, how many would you expect to have never been
married?
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You just computed the mean of a binomial distribution
More formally:
If X has the binomial distribution B(n, p), then
Mean of X =  is given by
Standard Deviation of X is given by
Note: These formulas are good only for ________ distributions and _____ for other
discrete random variables.
Example
Lie Detector Tests – given to truthful people have a probability of 0.2 of indicating a
person is lying. Suppose we ask 12 job applicants about thefts from previous employers
and that all 12 answer truthfully.
a. What is the probability that the test says at least 1 person is being deceptive?
b. What is the mean and standard deviation for the number of people classified as
deceptive?
c. What is the probability that the number classified as deceptive is less than the mean?
The Normal Approximation to Binomial Distributions
The formula for _______________ probabilities is harder to use when the number of
trials n ________________. When n is large, we can use Normal probability calculations
to approximate these binomial probabilities.
Example
About 12% of Americans are African American. Let X = the number of African
Americans in a random sample of 1500 Americans.
Then X has a _________________________ distribution.
a. What is the mean and standard deviation of the number of African Americans in the
sample?
b. What is the probability that the sample contains 165 or fewer African Americans?
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Normal Approximation for a Count
When X is a count having the B(n, p) distribution and n is large, then
X is approximately
So let’s try the Example, part b again…
Read Example 8.12 Page 527
Note:
Read Example 8.13 Page 528
Note:
Jumping ahead to Chapter 9, let’s look at:
Sampling Distribution of a Proportion
The count X of the number of successes is directly related to the proportion of successes
x
pˆ  .
n
So we have:
Mean of p̂ = mean of
X
=
n
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That is, we would expect the value of a sample proportion to be equal to the population
proportion, on average. This says that p̂ is an ______________ estimator for p…
Standard Deviation of p̂ = dev std of
X
n
That is, as n __________ the standard deviation __________, which is good, improves
the accuracy of the estimator overall.
We could convert any probability question about a proportion to that of a count, and viceversa.
If n is small, we would need to convert the question to a count and use the ___________
distribution to work it out. If n is large, we could convert the question to a _____and use
the _____________________for a count, OR use a related normal approximation for a
proportion (for large n).
Normal Approximation for a Proportion
Let p̂ be the sample proportion of successes in a random sample of size n
from a population with proportion of successes p.
if n is large, then p̂ is approximately
(rule: np  10 and n(1p)10)
Review the Binomial Distribution with the Calculator Pages 530-532.
Simulating Binomial Events
Read Example 8.14 Page 533
Let’s try Exercise 8.26 Page 534 Simulating defective switches
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Recap of Main ideas from Section 8.1
Scenario: We have some population of interest and a dichotomous characteristic (i.e. two
possible outcomes – success or failure) that we want to learn about.
Take a random sample of size n and compute
X = number of successes in the sample
In repeated sample of the same size n,
the values of X will ___________.
The distribution that describes how the
the X values vary is called the …
Take a random sample of size n and compute
= proportion of successes in the sample.
In repeated samples of the same size n,
The values of
will _____________.
Summary: In Section 8.1 the response is dichotomous (two outcomes, generically
__________and ________) and we have results for answering questions about counts or
proportions. These results require that we have a random sample (not just any sample).
If the _______________, we will be working with the binomial distribution and use the
calculator to find probabilities for binomial distributions for various values of n and p. If
the ____________________, we can rely on one of two (related) large sample normal
approximations and use Table A to find the approximate probabilities.
8.2 Geometric Distributions
Recall: If the random variable X is the count of successes in a ____________________,
then X has a ______________ distribution. Which means
 n = fixed number of observations
 n observations are _________________
 two outcomes – Bernoulli trials:
 probability of success on any outcome =
Similarly, if we have a set of Bernoulli trials and our variable of interest, X, is the number
of ____________ required to obtain the ___________ success, then we have a
_____________________________ and X has a _______________ distribution.
Note: the number of trials in a ________________________ is not __________ but is
the variable of interest.
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Read Example 8.16 Page 541
Note:
Using the calculator geometpdf (_________) Let’s try it with example 8.16.
Note:
A probability distribution for the geometric random variable never ends; that is, the
number of table entries is ___________.
Recall from algebra the general form for a geometric sequence:
Then the formula for the sum of an infinite geometric sequence is:
Example 8.17 Page 542
Note: distribution histogram is strongly skewed to the right with peak at 1.
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The Mean and Standard Deviation of a Geometric Random Variable
1
Mean or expected value:  x  p
Variance:
x12p
Var
p
Read Example 8.18 Page 545
Note:
If you are interested in the probability that it takes ________________ a certain number
of trials to achieve success then:
Example
Roll a die until a 5 is observed. The probability that it takes more than 7 rolls to observe
a 5:
Example (using geometpdf and geometcdf)
Two dice are rolled over and over until a sum of 7 or 11 is rolled. What is the probability
that it takes:
a. exactly 4 rolls
b. more than 2 rolls
c. at most 6 rolls
d. from 3 to 6 rolls
Let’s try Exercise 8.51 Page 552 Drawing marbles
20 red marbles, 10 blue marbles, and 5 white marbles in a jar. Select a marble without
looking, note the color and then replace the marble. We are interested in how many
marbles would have to be drawn in order to be sure you have a red marble.
a. binomial or geometric? Explain. Define X
b. P(X=2)
P(X  2)
P(X  2)
c.
d.
1
0