Chapter 5: Probability: What are the Chances?
Section 5.1
Randomness, Probability, and Simulation
The Practice of Statistics, 4th edition – For AP*
STARNES, YATES, MOORE
Chapter 5
Probability: What Are the Chances?
 5.1
Randomness, Probability, and Simulation
 5.2 Probability Rules
 5.3 Conditional Probability and Independence
Section 5.1
Randomness, Probability, and Simulation
Learning Objectives
After this section, you should be able to…
 DESCRIBE the idea of probability
 DESCRIBE myths about randomness
 DESIGN and PERFORM simulations
 Whose Book is This?
their next test in AP Statistics. When they go for a snack in
the kitchen, Tim’s three-year-old brother makes a tower using
their textbooks. Unfortunately, none of the students wrote his
name in the book, so when they leave each student takes one
of the books at random. When the students returned the
books at the end of the year and the clerk scanned their
barcodes, the students were surprised that none of the four
had their own book. How likely is it that none of the four
students ended up with the correct book?
 On four equally sized slips of paper, write the numbers 1, 2, 3, 4.
 Shuffle the papers and lay them down one at a time in a row. If
the number on the paper matches it’s position in the row (e.g.
paper 2 ends up in the second position), this represents a student
choosing his own book from the tower of textbooks. Count the
number of students who get the correct book.
 Repeat this several more times, recording the number of students
who get the correct book in each trial.
 Combine your results with your classmates and estimate how
often none of the four end up with their own book.
Randomness, Probability, and Simulation
 Suppose that 4 friends get together to study at Tim’s house for
 The Idea of Probability
Definition:
The probability of any outcome of a chance process is a
number between 0 (never occurs) and 1(always occurs) that
describes the proportion of times the outcome would occur in a
very long series of repetitions.
Randomness, Probability, and Simulation
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.
 Whose Book is This?
choose a book at random. The blue line is the correct probability of
0.375. As you can see, in the first 20 trials, there is quite a bit of
variability. However, after 500 trials, the proportion of times there
was no match is quite close to the actual value.
Randomness, Probability, and Simulation
 The graphs below show the short-run and long-run behavior of the
proportion of trials in which there are no matches when 4 students
extended warranty for a specific type of cell
phone? Suppose that 5% of these cell
phones under warranty will be returned and
the cost to replace the phone is $150. If the
company knew which phones would go bad,
it could charge $150 for these phones and
$0 for the rest. However, since the
company can’t know which phones will be
returned but knows that about 1 in every 20
will be returned, they should charge at least
150/20 = $7.50 for the extended warranty.
Randomness, Probability, and Simulation
 Extended Warranties
 How much should a company charge for an
 Myths about Randomness
The myth of short-run regularity:
Randomness, Probability, and Simulation
The idea of probability seems straightforward.
However, there are several myths of chance
behavior we must address.
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.
Roll a die 12 times and record the result of each roll. Which of the
following outcomes is more probable?
123456654321
154524336126
These outcomes are both equally (un)likely, even though the first set
of rolls has a more noticeable pattern.
 Ex: Red is Due!
Randomness, Probability, and Simulation
 Ex: Runs in Die Rolling
In casinos, there is often a large display next to every roulette table
showing the outcomes of the last several spins of the wheel. Since
the results of previous spins reveal nothing about the results of future
spins, why do the casinos pay for these displays? Because many
players use the previous results to determine what bets to make, even
though it won’t help them win. And as long as the players keep
making bets, the casino keeps making money.
 Simulation
Performing a Simulation
State: What is the question of interest about some chance process?
Plan: Describe how to use a chance device to imitate one repetition of the
process. Explain clearly how to identify the outcomes of the chance
process and what variable to measure.
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.
Randomness, Probability, and Simulation
The imitation of chance behavior, based on a
model that accurately reflects the situation, is
called a simulation.
 State: When four students mix up their AP Stats books,
what is the probability that when each student randomly
chooses a book, he doesn’t end up with his own?
 Plan: On four equally sized slips of paper, write the
numbers 1, 2, 3, 4. Shuffle the papers and lay them
down one at a time in a row. If the number on the paper
matches its position in the row (e.g., paper 2 ends up in
the second position), this represents a student choosing
his own book. Count the number of students who get
the correct book.
 Do: Do this process many times, recording the number
of students who get the correct book in each trial.
 Conclude: Out of 30 total repetitions, there were 11
times when none of the students ended up with their
own book. So the estimated probability is 11/30 =
36.7%.
Randomness, Probability, and Simulation
 Whose Book is This?
 Example: Golden Ticket Parking Lottery
Read the example on page 290.
What is the probability that a fair lottery would result in two winners from the
AP Statistics class?
We’ll use Table D to simulate choosing the golden ticket lottery winners. Label
theStats students 01-28 and the rest of the students 29-95. After finding two
winners, we will record whether both winners were members of the Statistics
Reading across row 139 in Table
class.
Students
Labels
D, look at pairs of digits until you
AP Statistics Class 01-28
see two different labels from 0195. Record whether or not both
Other
29-95
winners are members of the AP
Skip numbers from 96-00
Statistics Class.
55 | 58
89 | 94
04 | 70
70 | 84
10|98|43
56 | 35
69 | 34
48 | 39
45 | 17
X|X
X|X
✓|X
X|X
✓|Sk|X
X|X
X|X
X|X
X|✓
No
No
No
No
No
No
No
No
No
19 | 12
97|51|32
58 | 13
04 | 84
51 | 44
72 | 32
18 | 19
✓|✓
Sk|X|X
X|✓
✓|X
X|X
X|X
✓|✓
X|Sk|X
Sk|✓|✓
Yes
No
No
No
No
No
Yes
No
Yes
40|00|36 00|24|28
Based on 18 repetitions of our simulation, both winners came from the AP Statistics
class 3 times, so the probability is estimated as 16.67%.
 Example: NASCAR Cards and Cereal Boxes
Read the example on page 291.
What is the probability that it will take 23 or more boxes to get a
full set of 5 NASCAR collectible cards?
Driver
Label
Jeff Gordon
1
Dale Earnhardt, Jr.
2
Tony Stewart
3
Danica Patrick
4
Jimmie Johnson
5
Use randInt(1,5) to simulate buying one box of
cereal and looking at which card is inside. Keep
pressing Enter until we get all five of the labels
from 1 to 5. Record the number of boxes we
had to open.
3 5 2 1 5 2 3 5 4 9 boxes
4 3 5 3 5 1 1 1 5 3 1 5 4 5 2 15 boxes
5 5 5 2 4 1 2 1 5 3 10 boxes
We never had to buy more than 22 boxes to get the full set of cards in 50 repetitions of
our simulation. Our estimate of the probability that it takes 23 or more boxes to get a
full set is roughly 0.
Section 5.1
Randomness, Probability, and Simulation
Summary
In this section, we learned that…
 A chance process has outcomes that we cannot predict but have a
regular distribution in many distributions.
 The law of large numbers says the proportion of times that a
particular outcome occurs in many repetitions will approach a single
number.
 The long-term relative frequency of a chance outcome is its
probability between 0 (never occurs) and 1 (always occurs).
 Short-run regularity and the law of averages are myths of probability.
 A simulation is an imitation of chance behavior.
Looking Ahead…
In the next Section…
We’ll learn how to calculate probabilities using
probability rules.
We’ll learn about
 Probability models
 Basic rules of probability
 Two-way tables and probability
 Venn diagrams and probability