Transcript Performing a Simulation

CHAPTER 5
Probability: What Are
the Chances?
5.1
Randomness, Probability,
and Simulation
The Practice of Statistics, 5th Edition
Starnes, Tabor, Yates, Moore
Bedford Freeman Worth Publishers
Randomness, Probability, and Simulation
Learning Objectives
After this section, you should be able to:
 INTERPRET probability as a long-run relative frequency.
 USE simulation to MODEL chance behavior.
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Aim – How do we interpret the meaning of
probability?
H.W. – Read section 5.1 and do pages 300 – 302 #1, 3, 5,
8, 11, 17, 19, 21, 24 and 27
Do Now – Suppose you have a coin.
1. What is the probability of tossing a coin and getting
heads?
2. What is the probability of tossing a coin and getting
tails?
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The Idea of Probability
Chance behavior is unpredictable in the short run, but has a regular
and predictable pattern in the long run.
The law of large numbers says that if we observe more and more
repetitions of any chance process, the proportion of times that a
specific outcome occurs approaches a single value.
The probability of any outcome of a chance process is a number
between 0 and 1 that describes the proportion of times the
outcome would occur in a very long series of repetitions.
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Myths About Randomness
The idea of probability seems straightforward. However, there are
several myths of chance behavior we must address.
The myth of short-run regularity:
The idea of probability is that randomness is predictable in the long
run. Our intuition tries to tell us random phenomena should also be
predictable in the short run. However, probability does not allow us to
make short-run predictions.
The myth of the “law of averages”:
Probability tells us random behavior evens out in the long run. Future
outcomes are not affected by past behavior. That is, past outcomes
do not influence the likelihood of individual outcomes occurring in the
future.
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Check our understanding:
1. According to a recent study, the probability that a
randomly selected U.S. adult usually eats breakfast is
0.61.
a. Explain what probability of 0.61 means in this setting.
Solution:
• IF you ask a large sample of U.S. adults whether they
eat breakfast about 61% of them will answer yes (how
large is large???)
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Check our understanding continued
• Why doesn’t this probability say that if 100 U.S. adults
are chosen at random, exactly 61 of them usually eat
breakfast?
• Solution:
• The exact number of breakfast eaters will vary from
sample to sample.
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Simulation
The imitation of chance behavior, based on a model that accurately
reflects the situation, is called a simulation.
Performing a Simulation
State: Ask a question of interest about some chance process.
Plan: Describe how to use a chance device to imitate one
repetition of the process. Tell what you will record at the end of
each repetition.
Do: Perform many repetitions of the simulation.
Conclude: Use the results of your simulation to answer the
question of interest.
We can use physical devices, random numbers (e.g. Table D),
and technology to perform simulations.
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Example: Simulations with technology
In an attempt to increase sales, a breakfast cereal company decides to
offer a NASCAR promotion. Each box of cereal will contain a collectible
card featuring one of these NASCAR drivers: Jeff Gordon, Dale
Earnhardt, Jr., Tony Stewart, Danica Patrick, or Jimmie Johnson.
The company says that each of the 5 cards is equally likely to appear in
any box of cereal.
A NASCAR fan decides to keep buying boxes of the cereal until she has
all 5 drivers’ cards. She is surprised when it takes her 23 boxes to get the
full set of cards. Should she be surprised?
Problem: What is the probability that it will take 23 or more boxes to get a full
set of 5 NASCAR collectible cards?
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Example: Simulations with technology
Plan: We need five numbers to represent the five possible cards.
Let’s let 1 = Jeff Gordon,
2 = Dale Earnhardt, Jr.,
3 = Tony Stewart,
4 = Danica Patrick, and
5 = Jimmie Johnson.
We’ll use randInt(1,5) to simulate buying one box of cereal and looking at
which card is inside.
Because we want a full set of cards, we’ll keep pressing Enter until we get
all five of the labels from 1 to 5. We’ll record the number of boxes that we
had to open.
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Example: Simulations with technology
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5
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5
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5
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9 boxes
2 2 4 4 5 3
16 boxes
10 boxes
1 5 4 5 2
15 boxes
2 2 3 3 3 2 3 3 4 2 2 5
22 boxes
Conclude: We never had to buy more than 22 boxes to get the full set of
NASCAR drivers’ cards in 50 repetitions of our simulation. So our estimate
of the probability that it takes 23 or more boxes to get a full set is roughly
0. The NASCAR fan should be surprised about how many boxes she had
to buy.
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Using a table for simulation:
• Textbook:
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Another example:
• Suppose I want to choose a simple random sample of size 6 from a
group of 60 seniors and 30 juniors. To do this, I write each person’s
name on an equally-sized piece of paper and mix the papers in a
large grocery bag. Just as I am about to select the first name, a
thoughtful student suggests that I should stratify by class. I agree,
and we decide it would be appropriate to select 4 seniors and 2
juniors. However, because I have already mixed up the names, I
don’t want to have to separate them all again. Instead, I will select
names one at a time from the bag until I get 4 seniors and 2 juniors.
Design and carry out a simulation using Table D to estimate the
probability that you must draw 10 or more names to get 4 seniors
and 2 juniors.
• State: What is the probability that it takes 10 or more selections to
get 4 seniors and 2 juniors?
• What would a possible simulation look like?
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Another example continued:
• Plan: Using pairs of digits from Table D:
• We’ll label the 60 seniors 01–60 and the 30 juniors 61–
90. Numbers 00 and 91–99 will be skipped. Moving left
to right across a row, we’ll look at pairs of digits until we
have 4 different labels from 01–60 and 2 different labels
from 61–90. Then we will count how many different
labels from 01–90 we looked at.
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Continued:
• Do: Here is an example of one repetition, using line 101 from Table
D:
• 19 (senior) 22 (senior) 39 (senior) 50 (senior) 34 (senior) 05 (senior)
75 (junior) 62 (junior)
• In this
trial, it took exactly 8 selectionsDot
toPlot
get at least 4 seniors and
Collection 2
at least 2 juniors.
• Here are the results of 50 trials:
6
7
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9 10 11 12 13 14
NumSel
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Conclusion:
• Conclude: In the simulation, 11 of the 50 trials required 10 or more
selections to get 4 seniors and 2 juniors, so the probability that it
takes 10 or more selections is approximately 0.22.
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A.P. Exam tip
• When students make conclusions, they often lose credit for
suggesting that a claim is definitely true or saying that the evidence
proves that a claim is incorrect. If the conclusion is based on a
probability, students should always acknowledge that they can’t be
certain that their conclusion is correct. A better response would be to
say that there is convincing evidence (or there isn’t convincing
evidence) to support a particular claim.
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Randomness, Probability, and Simulation
Section Summary
In this section, we learned how to…
 INTERPRET probability as a long-run relative frequency.
 USE simulation to MODEL chance behavior.
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