Download AP Stat 5.1 PP

CHAPTER 5
Probability: What Are the
Chances?
5.1: Randomness, Probability, and
Simulation
Complete 5.1 HW #1 tonight
The Practice of Statistics, 5th Edition
Starnes, Tabor, Yates, Moore
Bedford Freeman Worth Publishers
Do Now Activity
As a special promotion for its 20-ounce bottle of soda, a
soft drink printed a message on the inside of each bottle
cap. Some of the caps said, “Please try again!” while
others said, “You’re a winner!” The company advertised
the promotion with the slogan “1 in 6 wins a prize.” The
prize is a free 20-ounce bottle of soda. Seven friends
each buy one 20-ounce bottle at a local convenience
store. The store clerk is surprised when three of them
win a prize. The store owner is concerned about losing
money from giving away too many free sodas. She
wonders if this group of friends is just lucky or if the
company’s 1-in-6 claim is inaccurate.
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Do Now Activity
In this activity, you will perform a simulation to help
answer this question. We can model the status of an
individual bottle with a six-sided die: let 1-5 represent
“Please try again!” and 6 represent “You’re a winner!”
1. Roll your die seven times to imitate the process of
the seven friends buying their sodas. How many of
them won a prize?
2. Come up and record the data for a class dotplot.
3. Everyone repeat until we have 40 repetitions of the
simulation.
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Dotplot and results
0
1
2
3
4
5
6
7
Number of winners in a group of seven
4. Discuss results. What percent of the time did the
friends come away with three or more prizes, just by
chance? Does it seem plausible that the company is
telling the truth, but that the seven friends just got
lucky? Explain.
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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.
Probability describes what happens in very many trials, and
we must actually observe many trials to pin down
probability.
Probability gives us a language to describe the long-term
regularity of random behavior.
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The Idea of Probability
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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Check your understanding
1. According to the Book of Odds Web site, the probability
that a randomly selected U.S. adult usually eats breakfast
is 0.61.
a) Explain what the probability 0.61 means in this setting.
b) Why doesn’t this probability say that if 100 U.S. adults are
chosen at random, exactly 61 of them usually eat breakfast?
2. Probability is a measure of how likely an outcome is to
occur. Match one of the probabilities that follow with each
statement.
0 0.01 0.3
0.6
0.99 1
a) The outcome is impossible. It can never occur.
b) The outcome is certain. It will occur on every trial.
c) This outcome is very unlikely, but it will occur once in a while
in a long sequence of trials.
d) This outcome will occur more often than not.
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Myths About Randomness
The idea of probability seems straightforward. However, there
are several myths of chance behavior we must address.
1. Toss a coin six times and record heads (H) or tails (T) on
each toss. Which of the following outcomes is more probable?
HTHTTH
TTTHHH
– Both are equally likely.
– Heads and tails are equally probable says only that about
half of a very long sequence of tosses will be heads. It
doesn’t say that heads and tails must come close to
alternating in the short run.
– The coin has no memory. It doesn’t know what past
outcomes were, and it can’t try to create a balanced
sequence.
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Myths About Randomness
The idea of probability seems straightforward. However, there
are several myths of chance behavior we must address.
2. A couple has had 7 children, all girls, they decided to have
another child because they must be due for a boy. Are they
more likely to have a boy than a girl?
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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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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 (e.g. slips of paper, die), 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
3 5 2 1 5 2 3 5 4
9
boxes
5 1 2 5 1 4 1 4 1 2 2 2 4 4 5 3
16 boxes
5 5 5 2 4 1 2 1 5 3
10
boxes
4 3 5 3 5 1 1 1 5 3 1 5 4 5 2
15 boxes
3 3 2 2 1 2 4 3 3 4 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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Check your understanding
In the NASCAR and breakfast cereal example, what if the
cereal company decided to make it harder to get some
drivers’ cards than others? For instance, suppose the
chance that each card appears in a box of cereal is Jeff
Gordon, 10%; Dale Earnhardt, Jr., 30%; Tony Stewart,
20%; Danica Patrick, 25%; and Jimmie Johnson, 15%.
How would you modify the simulation in the example to
estimate the chance that a fan would have to buy 23 or
more boxes to get the full set?
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