Chapter 16
Probability Models
Copyright © 2015, 2010, 2007 Pearson Education, Inc.
Chapter 16, Slide
1-1 1
Bernoulli Trials
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The basis for the probability models we will examine in
this chapter is the Bernoulli trial.
We have Bernoulli trials if:
 there are two possible outcomes (success and failure).
 the probability of success, p, is constant (fixed).
 the trials are independent.
Copyright © 2015, 2010, 2007 Pearson Education, Inc.
Chapter 16, Slide
1-2 2
The Geometric Model
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A single Bernoulli trial is usually not all that interesting.
A Geometric probability model tells us the probability for a
random variable that counts the number of Bernoulli trials
until the first success.
Geometric models are completely specified by one
parameter, p, the probability of success, and are denoted
Geom(p).
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Chapter 16, Slide
1-3 3
The Geometric Model (cont.)
Geometric probability model for Bernoulli trials:
Geom(p)
p = probability of success
q = 1 – p = probability of failure
X = number of trials until the first success occurs
x-1
P(X = x) = q p
1
E(X) = m =
p
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Chapter 16, Slide
1-4 4
Independence
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One of the important requirements for Bernoulli trials is
that the trials be independent.
When we don’t have an infinite population, the trials are
not independent. But, there is a rule that allows us to
pretend we have independent trials:
 The 10% condition: Bernoulli trials must be
independent. If that assumption is violated, it is still
okay to proceed as long as the sample is smaller than
10% of the population.
Copyright © 2015, 2010, 2007 Pearson Education, Inc.
Chapter 16, Slide
1-5 5
The Binomial Model
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A Binomial model tells us the probability for a
random variable that counts the number of
successes in a fixed number of Bernoulli
trials.
Two parameters define the Binomial model: n,
the number of trials; and, p, the probability of
success. We denote this Binom(n, p).
Copyright © 2015, 2010, 2007 Pearson Education, Inc.
Chapter 16, Slide
1-6 6
The Binomial Model (cont.)

In n trials, there are
n!
n Ck =
k !(n - k )!
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ways to have k successes.
Read nCk as “n choose k.”
nCk can also be written
𝑛
𝑘
Note: n!= n ´ (n -1) ´ ...´ 2 ´1, and n! is read
as “n factorial.”
Copyright © 2015, 2010, 2007 Pearson Education, Inc.
Chapter 16, Slide
1-7 7
The Binomial Model (cont.)
Binomial probability model for Bernoulli trials:
Binom(n,p)
n = number of trials
p = probability of success
q = 1 – p = probability of failure
X = # of successes in n trials
x
n–x
P(X = x) = nCx p q
m = np
s = npq
Copyright © 2015, 2010, 2007 Pearson Education, Inc.
Chapter 16, Slide
1-8 8
The Normal Model to the Rescue!*
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
When dealing with a large number of trials in a
Binomial situation, making direct calculations of
the probabilities becomes tedious (or outright
impossible).
Fortunately, the Normal model comes to the
rescue…
Copyright © 2015, 2010, 2007 Pearson Education, Inc.
Chapter 16, Slide
1-9 9
The Normal Model to the Rescue (cont.)

As long as the Success/Failure Condition holds,
we can use the Normal model to approximate
Binomial probabilities.
 Success/failure condition: A Binomial model is
approximately Normal if we expect at least 10
successes and 10 failures:
np ≥ 10 and nq ≥ 10
See example on pages 425-426 in your text
book.
Copyright © 2015, 2010, 2007 Pearson Education, Inc.
Chapter 16, Slide
1-10 10
Continuous Random Variables
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When we use the Normal model to approximate
the Binomial model, we are using a continuous
random variable to approximate a discrete
random variable.
So, when we use the Normal model, we no
longer calculate the probability that the random
variable equals a particular value, but only that it
lies between two values.
Copyright © 2015, 2010, 2007 Pearson Education, Inc.
Chapter 16, Slide
1-11 11
What Can Go Wrong?
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Be sure you have Bernoulli trials.
 You need two outcomes per trial, a constant
probability of success, and independence.
 Remember that the 10% Condition provides a
reasonable substitute for independence.
Don’t confuse Geometric and Binomial models.
Don’t use the Normal approximation with small n.
 You need at least 10 successes and 10
failures to use the Normal approximation.
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Chapter 16, Slide
1-12 12
What have we learned?
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Bernoulli trials show up in lots of places.
Depending on the random variable of interest, we
might be dealing with a
 Geometric model
 Binomial model
 Normal model
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Chapter 16, Slide
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What have we learned? (cont.)
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Geometric model
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Binomial model
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When we’re interested in the number of Bernoulli
trials until the next success.
When we’re interested in the number of successes
in a certain number of Bernoulli trials.
Normal model

To approximate a Binomial model when we expect
at least 10 successes and 10 failures.
Copyright © 2015, 2010, 2007 Pearson Education, Inc.
Chapter 16, Slide
1-14 14
AP Tips

The AP rubrics usually have three requirements
for probability problems:
 Name of distribution
 Identification of correct parameters
 Correct calculation
Copyright © 2015, 2010, 2007 Pearson Education, Inc.
Chapter 16, Slide
1-15 15
AP Tips
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
The Name can be identified with
 words (binomial) or with
x n–x
 the proper formula (P(X = x) = nCx p q
)
 (Your teacher will probably ask you to write both
and that’s a good thing!)
The parameters should use standard notation:
 n = 10, p = 0.7 or
 μ = 65”, σ = 3” (for a normal problem)
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Chapter 16, Slide
1-16 16
Examples
A new sales gimmick has 30% of M&M’s covered with
speckles. These new candies are mixed randomly with
normal M&M’s as they are put in bags for distribution
and sale. You buy a bag a remove candies one at a
time looking for speckles.
1)
What’s the probability that the first speckled M&M
we see is the fourth on removed from the bag? The
10th?
2)
What’s the probability that the first speckled M&M is
one of the first three?
3)
How many M&M’s do we expect to have to check to
find a speckled one?
4)
What’s the probability that we will find 2 speckled
M&M’s in a handful of 5?
Copyright © 2015, 2010, 2007 Pearson Education, Inc.
Chapter 16, Slide
1-17 17
4)
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