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
P(X = x) = q
x-1
p
E
(
X
) 1
p
q p
2 Slide 1- 4
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 4 th shot?
• What is the probability you will make your first basket before or on the 4 th 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.
8
The Binomial Model
• In
n
trials, there are
n C x
n
!
x
!
ways to have
x
successes. – Read
n C x
as “ combination.
n
choose
x
,” and is called a • Note:
n
! =
n
x (
n – 1
) x read as “
n
factorial.”
…
x
2
x
1
, and
n
! is 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
x
) (
n
n
!
np x p q
npq
10
Example
A report from the Secretary of Health and Human Services stated that 70% of single vehicle 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. 11
7 8 9 10
x
0 1 2 3 4 5 6
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.
16
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’s eyes? 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
.
19
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)
x
e
x
!
x
20
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 23
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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