Transcript Slide 1

Chapter 17
Probability Models
Copyright © 2009 Pearson Education, Inc.
Objectives:
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The student will be able to:
 Tell if a situation involves Bernoulli trials.
 Know the appropriate conditions for using a
Binomial or Normal model.
 Find and interpret in context the mean and
standard deviation of a Binomial model.
 Calculate binomial probabilities, perhaps with a
Normal model.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 3
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.
 the trials are independent.
Examples:
 Flipping a coin (where heads is success), rolling a die
(where getting a “6” is success), throwing free throws
in a basketball game, drawing a card from a deck of
cards with replacement (where drawing an Ace is
success)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 4
Do we have Bernoulli Trials?
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You are rolling 5 dice and need to get at least two 6’s to win the
game
We record the eye colors found in a group of 500 people
A city council of 11 Republicans and 8 Democrats picks a
committee of 4 at random. What is the probability that they
choose all Democrats?
A 2002 Rutgers University study found that 74% of high school
students have cheated on a test at least once. Your local high
school principle conducts a survey and gets responses that
admit to cheating from 322 of 481 students.
How likely is it that in a group of 120 the majority may have
type A blood, given that Type A is found in 43% of the
population?
Copyright © 2009 Pearson Education, Inc.
Slide 1- 5
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.
 Example: lets draw cards from a standard deck with
replacement and consider drawing a heart “success.”
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Do we have Bernoulli trials? Would we have Bernoulli trials if
we were drawing without replacement?
What is the probability p of success? What is the probability q
of failure?
What is the probability that the first heart is the 3rd card drawn?
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i.e. first success occurs on trial 3.
Geometric models are completely specified by one
parameter, p, the probability of success, and are denoted
Geom(p).
Slide 1- 6
Copyright © 2009 Pearson Education, Inc.
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
In our example P(X=3) = (39/52)2(13/52)
1
E(X)   
p
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
q
p2
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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.
 Example: If success is drawing a heart (drawing with
replacement), what is the probability that if we draw
and replace 3 cards that we drew exactly one heart?
Two parameters define the Binomial model: n, the number
of trials; and, p, the probability of success. We denote this
Binom(n, p).
 Example: If we flip a coin 6 times what is the
probability of getting heads exactly 3 times?
Copyright © 2009 Pearson Education, Inc.
Slide 1- 9
The Binomial Model (cont.)
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In n trials, there are
n!
n Ck 
k ! n  k !
ways to have k successes.
 Read nCk as “n choose k,” and is called a
combination.
 Example: How many ways are there to roll a
die five times and roll a 6 three of those times?
Note: n! = n x (n – 1) x … x 2 x 1, and n! is read
as “n factorial.”
Copyright © 2009 Pearson Education, Inc.
Slide 1- 10
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 = number of successes in n trials
n!
 n  x n x
n
P( X  x)    p q where  
 x
 x  x !(n  x)!
  np
Copyright © 2009 Pearson Education, Inc.
  npq
Slide 1- 11
Using the TI
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Suppose a light bulb company has a 20% defective rate.
Consider taking a sample of 6 bulbs.
1) What is the probability of getting exactly 1 defective bulb in
that group of 6? (Even though a defect isn't pleasant at times, it
is considered a success in this experiment since that is where
our focus is!)
If we computed this probability long hand we would do
1
5
6C1(.20) (.80)
On the calculator:
 2nd Distr...(#0) for binompdf(6, .2, 1) ... enter to get .393216
binompdf gives you the probability at a particular x. The pdf
must be followed by n (total number of trials), p (probability
of a success), x (number of successes you are interested
in)
 binomcdf, which we use next, will compute cumulative
probabilities.
Slide 1- 12
Copyright © 2009 Pearson Education, Inc.
Using the TI
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2) What is the probability of getting at most 2 defective light bulbs?
This means P(0) + P(1) + P(2)
OR
use the cumulative binomial button:
 2nd Distr...#A for binomcdf(6, .2, 2) ...enter to get .90112
3) What is the probability of getting at least two defective light bulbs?
 At least two means two or more which is the same as adding the
probabilities of 2 to 3 to 4 etc...
 OR 1 minus the complement of "at least two" which is 1 minus the
cdf to 1
 1 - binomcdf (6,.2,1) = .34464
4) What is the probability of getting from two to four defective light
bulbs? You could do the pdf for 2 + pdf for 3 + pdf for 4 or be a little
creative and do
 binmocdf(6,.2,4) - binomcdf(6, .2,1) = .34304
Copyright © 2009 Pearson Education, Inc.
Slide 1- 13
Using StatCrunch
To use StatCrunch to calculate Binomial Probabilities
(or to view the binomial probability histogram) go to
 Stat -> Calculators -> Binomial
 Enter the appropriate n, p, and Prob statement. Then
click "Calculate"
 For example, if you want to compute the
probability of observing at least one "6" in 5 rolls
of the die,
 n = 5
 p = 0.1667
 Prob (X=>1) = 0.598
Copyright © 2009 Pearson Education, Inc.
Slide 1- 14
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20) An Olympic archer is able to hit the bull’s-eye 80%
of the time. Assume each shot is independent of the
others. If she shoots 6 arrows, what’s the probability of
the following
rd arrow (note: this
 Her first bull’s-eye comes on the 3
uses the Geometric not the Binomial Distribution)
 She misses the bull’s-eye at least once
 Her first bull’s-eye comes on the fourth or fifth arrow
(Geometric)
 She gets exactly 4 bull’s-eyes
 She gets at least 4 bull’s-eyes
 She gets at most 4 bull’s-eyes
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Slide 1- 15
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17) If you flip a fair coin 100 times
 Intuitively how many heads do you expect?
 Use the formula for expected value to verify
your intuition
18) An American roulette wheel has 38 slots, of
which 18 are red, 18 are black, and 2 are green.
If you spin the wheel 38 times
 Intuitively how many times do you expect the
ball to land in a green slot?
 Use the formula for expected value to verify
your intuition
Copyright © 2009 Pearson Education, Inc.
Slide 1- 16
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22a, b Consider the same archer
 How many Bull’s-eyes do you expect her to get?
 With what standard deviation?
Suppose our archer shoots 10 arros
 Find the mean and standard deviation of the number
of bull’s-eyes you may get
 What’s the probability that she never misses?
 What the probability that there are no more than 8
bull’s-eyes
 What’s the probability that there are exactly 8 bull’seyes
 What’s the probability that she hits the bull’e-eye
more often than she misses
Copyright © 2009 Pearson Education, Inc.
Slide 1- 17
Practice
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6) Suppose 75% of all drivers always wear their
seatbelts. Lets investigate how many of the drivers
might be belted among six cars waiting at a traffic light.
 Describe how you’ll simulate the number of seatbelt
wearing drivers among the six cars
 Run 30+ trials
 Based on the simulation estimate the probabilities
that there are exactly no belted drivers, one, two,
three, etc.
 Calculate the actual probability model
 Compare the distribution of outcomes in the
simulation to the actual model
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Slide 1- 18
Recall our election example from
chapter 11
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If your candidate is favored by approximately
53% of the population, but only 100 people vote,
what is the probability that your candidate wins?
 In other words, your candidate needs at least
51 votes of 100 votes. Assume each voter is
independent.
 Do we have Bernoulli trials?
 What is the probability of “success” for a trial?
 What the probability that at least 51 of 100
voters vote for your candidate?
Copyright © 2009 Pearson Education, Inc.
Slide 1- 19
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 © 2009 Pearson Education, Inc.
Slide 1- 20
The Normal Model to the Rescue (cont.)
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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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Slide 1- 21
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.
Ex. For our election example:
 μ = np = 100*.53 =53
 σ = √(npq) = √(100*.53*.47) =4.99
 P(at least 51 votes) ~ Normalcdf(51,100, 53, 4.99)
Copyright © 2009 Pearson Education, Inc.
Slide 1- 22
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.
Copyright © 2009 Pearson Education, Inc.
Slide 1- 23
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 successes in a certain
number of Bernoulli trials.
Normal model
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When we’re interested in the number of Bernoulli trials until the
next success.
To approximate a Binomial model when we expect at least 10
successes and 10 failures.
Poisson model
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To approximate a Binomial model when the probability of
success, p, is very small and the number of trials, n, is very
large.
Copyright © 2009 Pearson Education, Inc.
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