Transcript Document

Chapter 17
Probability Models
1
Bernoulli Trials
• 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.
– the trials are independent.
2
The Geometric Model
• 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).
Slide 1- 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)   
p
Slide 1- 4

q
p2
Geometric Model - Example
• Suppose you are shooting free throws. After
rigorous data collection and calculations, you
find that your probability of making a free throw
to be 0.3.
• Assume that you meet the conditions for
Bernoulli trials.
• What is the probability you will make your first
basket on the 4th shot?
• What is the probability you will make your first
basket before or on the 4th shot?
5
Let’s take a POP quiz!
Suppose you come to class to find you are
having a pop quiz. The quiz has only four
multiple choice questions. You have not
had time to prepare for this quiz so you
are completely guessing for each
question. There are a total of 5 choices
(one of which is right) for each question.
What is the probability that you get exactly
three correct?
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The Binomial Model
• A Binomial model tells us the probability
for a random variable that counts the
number of successes in a fixed number of
trials.
• Two parameters define the Binomial
model: n, the number of trials; and, p, the
probability of success. We denote this
Binom(n, p).
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Is this a situation for the Binomial?
• Determining whether each of 3000 heart
pacemakers is acceptable or defective.
• Surveying people and asking them what
they think of the current president.
• Spinning the roulette wheel 12 times and
finding the number of times that the
outcome is an odd number.
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The Binomial Model
• In n trials, there are
n!
n Cx 
x ! n  x !
ways to have x successes.
– Read nCx as “n choose x,” and is called a
combination.
• Note: n! = n x (n – 1) x … x 2 x 1, and n! is
read as “n factorial.”
9
The Binomial Model (cont.)
n = number of trials
p = probability of success
q = 1 – p = probability of failure
x = number of successes in n trials
n!
x n x
P( X  x) 
pq
(n  x)! x !
  np
  npq
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Example
A report from the Secretary of Health and
Human Services stated that 70% of singlevehicle traffic fatalities that occur at night
on weekends involve an intoxicated driver.
If a sample of 10 single-vehicle traffic
fatalities that occur at night on a weekend
is selected, find the probability that exactly
5 involve a driver that is intoxicated.
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x
0
1
2
3
4
5
6
7
8
9
10
P(x)
0.000005
0.000138
0.001447
0.009002
0.036757
0.102919
0.200121
0.266828
0.233474
0.121061
0.028248
Using table generated in
MINITAB
P(x=5) = 0.102919
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How about the probability of at least 8
involving an intoxicated driver?
13
You try this one!
The participants in a television quiz show
are picked from a large pool of applicants
with approximately equal numbers of
men and women. Among the last 10
participants there have been only 2
women. If participants are picked
randomly, what is the probability of getting
2 or fewer women when 10 people are
picked?
14
The Normal Model to the Rescue!
• 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…
15
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.
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Continuous Random Variables
• 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.
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Example:
An Olympic archer is able to hit the bull’s-eye 80% of the
time. Assume that each shot it independent of the
others. She will be shooting 200 arrows in a large
competition.
a. What are the mean and standard deviation for the
number of bull’s-eyes she might get?
b. Is the normal model appropriate here?
c. Use the 68-95-99.7% Rule to describe the distribution of
the number of bull’s-eye she might get.
d. Would you be surprised if she only made 140 bull’seyes? Explain.
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The Poisson Model
• The Poisson probability model was
originally derived to approximate the
Binomial model when the probability of
success, p, is very small and the number
of trials, n, is very large.
• The parameter for the Poisson model is λ.
To approximate a Binomial model with a
Poisson model, just make their means
match: λ = np.
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The Poisson Model (cont.)
Poisson probability model for successes:
Poisson(λ)
λ = mean number of successes
X = number of successes
e is an important mathematical constant
(approximately 2.71828)

e 
P  X  x 
x!
EX   
x
SD  X   
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The Poisson Model (cont.)
• Although it was originally an approximation
to the Binomial, the Poisson model is also
used directly to model the probability of
the occurrence of events for a variety of
phenomena.
– It’s a good model to consider whenever your
data consist of counts of occurrences.
– It requires only that the events be
independent and that the mean number of
occurrences stays constant.
21
What Can Go Wrong?
• 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.
22
What have we learned?
• 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
– Poisson model
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What have we learned? (cont.)
– Geometric model
• When we’re interested in the number of Bernoulli trials until
the next success.
– Binomial model
• 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.
– Poisson model
• To approximate a Binomial model when the probability of
success, p, is very small and the number of trials, n, is very
large.
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