Download Chapter 17 - Geneva Area City Schools

Chapter 17
Probability Models
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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.

Ex: shooting a free throw, flipping tails on a coin
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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.
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Example (p. 401 #2)

Do these situations involve Bernoulli trials? Explain.
 a) You are rolling 5 dice and need to get at least two 6’s
to win the game.

b) We record the distribution of eye colors found in a
group of 500 people.

c) A manufacturer recalls a doll because about 3%
have buttons that are not properly attached. Customers
return 37 of these dolls to the local toy store. Is the
manufacturer likely to find any dangerous buttons?
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Example (p. 401 #2)

Do these situations involve Bernoulli trials? Explain.
 d) A city council of 11 Republicans and 8 Democrats
picks a committee of 4 at random. What’s the
probability they choose all Democrats?

e) A 2002 Rutgers University study found that 74% of
high school students have cheated on a test at least
once. Your local high school principal conducts a
survey in homerooms and gets responses that admit to
cheating from 322 of the 481 students.
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Answers to Example (p. 401 #2)
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The Geometric Model
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
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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).

Ex: number of free throws shot until one is made
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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
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
q
p2
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Example (p. 402 #10, 12)

Suppose a computer chip manufacturer rejects 2% of the chips
produced because they fail presale testing.
 10a) What’s the probability that the fifth chip you test is the first bad
one you find?


10b) What’s the probability you find a bad one within the first 10
you examine?
12) How many do you expect to test before finding a bad one?
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Answers to Example (p. 402 #10, 12)
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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 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).
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The Binomial Model (cont.)

In n trials, there are
n!
n Ck 
k ! n  k !
ways to have k successes.
 Read nCk as “n choose k.”


Order does not matter
Note: n! = n  (n – 1)  …  2  1, and n! is read
as “n factorial.”
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The Binomial Model (cont.)
n!
n Ck 
k ! n  k !

Example:

How many ways can you pick 3 people from a
team of 10?
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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
  np
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  npq
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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…
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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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What have we learned? (cont.)

Geometric model


Binomial model


When we’re interested in the number of Bernoulli
trials until the first 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.
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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.
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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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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
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Example (p. 401 #8)

A Department of Transportation report about air
travel found that airlines misplace about 5 bags
per 1000 passengers. Suppose you are traveling
with a group of people who have checked 22
pieces of luggage on your flight. Can you
consider the fate of these bags to be Bernoulli
trials? Explain.
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Example (p. 402 #18)

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 each of the
following results?
 a) Her first bull’s-eye comes on the third arrow.

b) She misses the bull’s-eye at least once.

c) Her first bull’s-eye comes on the fourth or fifth arrow.
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Example (p. 402 #18)

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 each of the following results?
 d) She gets exactly 4 bull’s-eyes.

e) She gets at least 4 bull’s-eyes.

f) She gets at most 4 bull’s-eyes.
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Homework

Chapter 17 Homework: p. 401 # 1, 7, 9, 11, 17
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Example (p. 402 #20)

An Olympic archer is able to hit the bull’s-eye 80% of the
time. Assume each shot is independent of the others.
She shoots 6 arrows.
 a) How many bull’s-eyes do you expect her to get?


b) With what standard deviation?
c) If she keeps shooting arrows until she hits the bull’seye, how long do you expect it will take?
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Example (p. 402 #22)

An Olympic archer is able to hit the bull’s-eye 80% of the
time. Assume each shot is independent of the others. She
shoots 10 arrows.
 a) Find the mean and standard deviation of the number of
bull’s-eyes she may get.

b) What’s the probability that

i) she never misses?

ii) there are no more than 8 bull’s-eyes?

iii) there are exactly 8 bull’s-eyes?

iv) she hits the bull’s-eye more often than she misses?
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Example (p. 403 #28)

An Olympic archer is able to hit the bull’s-eye 80% of the time. Assume
each shot is independent of the others. She shoots 200 arrows in a large
competition.
 a) What are the mean and standard deviation of the number of bull’seyes she might get?

b) Is a Normal model appropriate here? Explain.

c) Use the 68-95-99.7 Rule to describe the distribution of the number of
bull’s-eyes she may get.

d) What’s the probability she makes only 140 bull’s-eyes?
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Example (p. 403 #34)

Shortly after the introduction of the euro coin in
Belgium, newspapers around the world published
articles claiming the coin is biased. The stories
were based on reports that someone had spun
the coin 250 times and gotten 140 heads—that’s
56% heads. Do you think this is evidence that
spinning a euro is unfair? Explain.
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Homework
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Chapter 17 Homework: p. 401 #19, 21, 25, 27,
33
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