Transcript File

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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
+ 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
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.
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Check Your Understanding
Example 1
Probability is a measure of how likely an outcome is to occur. Match
one of the probabilities that follow with each statement. Be prepared
to defend your answer. {0, 0.01, 0.3, 0.6, 0.99, 1}
a)
This outcome is impossible. It can never occur.
b)
This outcome is certain. It will occur on every trial.
c)
This outcome is very unlikely, but it will ocurr once in a while in a
long sequence of trials.
d)
This outcome will occur more often than not.
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Solution

0 – never occur

1 – always occur

0.01 – approximately 1%

0.6 – approximately 60% will occur more than not occur,
whereas .99 means it happens ALMOST every time.
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
Example 2
4 students are studying together.
While they are getting a snack in the
kitchen, someone’s younger brother
comes in and mixes up all the
textbooks. Each student takes a book
at random. The graphs below show
the short-run and long-run behavior of
the proportion of trials in which there
are no matches when four students
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.
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.
Randomness, Probability, and Simulation
Chance behavior is unpredictable in the short run, but has a regular and
predictable pattern in the long run.
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 The
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Example 3

How much should a company charge for an 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. What is the minimum amount the company
should charge for the extended warranty?
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Solution

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.
+ Myths about Randomness
 The
idea about probability is that randomness is
predictable in the long run. Unfortunately, our
intuition about randomness tries to tell us that
random phenomena should also be predication in
the short run. When they aren’t, we look for some
explanation other than chance variation.
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Example 4
123456654321
or
154524336126
Randomness, Probability, and Simulation
Roll a die 12 times and record the result of each roll. Which of the
following outcomes is more probable?
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4 Solution
These outcomes are both equally (un)likely, even though the first set
of rolls has a more noticeable pattern.
Randomness, Probability, and Simulation
 Example
Example 5
+Suppose
that a basketball announcer suggests a certain player is streaky.

That is, the announcer believes that if the player makes a shot, then the
player is more likely to make his next shot. As evidence, the announcer
points to a recent game where the player took 30 shots and had a streak of
7 made shots in a row. Is this convincing evidence of streakiness, or could
it have occurred simply by chance? Assuming that this player makes 50%
of his shots and the results of a shot don’t depend on previous shots, how
likely is it for the player to have a streak of 7 or more made shots in a row?

Using your calculator, let 1 represent a made shot and 0 represent a missed
shot.

Calculate a random integer (either 0 or 1) 30 times, writing down the
outcome after each calculation. Record the length of the longest streak of
made shots.

Combine your results with those of your classmates. In what proportion of
the trials did the player have a streak of at least 7 in a row?
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STATE

How likely is it for the player to have a streak of 7 or more shots
in a row?
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PLAN

Use the random number generator on the calculator to
generate 30 random shots.

0 represents a missed shot. 1 represents a shot made.

Record the outcome of each trial.

Record the longest streaks on a dotplot.
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DO

Perform this twice.
_________________________________________________________________________________
2
3
4
5
6
7
8
9
10
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CONCLUDE

The player had a streak of 7 or more made shots in ____ of
the ____ simulated games (_____%).
Randomness, Probability, and Simulation
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?
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Example 6
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Solution
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.
+ Section 5.1
Randomness, Probability, and Simulation
Learning Objectives
After this section, you should be able to…

DESIGN and PERFORM simulations
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 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.
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IMPORTANT
When making conclusions, be careful to say that there is either
convincing evidence (or there isn’t convincing evidence) to
support a particular claim.
DO NOT say that a claim is definitely true or that the evidence
proves that a claim is incorrect.
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Example 1

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.
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STATE

What is the probability that it takes 10 or more selections to get
4 seniors and 2 juniors?
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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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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 selections to get at least 4 seniors
and at least
2 juniors.
Here are the results ofDot50
Plottrials:
Collection
2
6
7
8
9
10 11
NumSe l
12
13
14
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CONCLUDE
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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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Example 2

At their annual picnic, 18 students in the mathematics/statistics
department at a university decide to play a softball game.
Twelve of the 18 students are math majors and 6 are stats
majors. To divide into two teams of 9, one of the professors put
all the players’ names into a hat and drew out 9 players to form
one team, with the remaining 9 players forming the other team.
The players were surprised when one team was made up
entirely of math majors.

Is it possible that the names weren’t adequately mixed in the
hat, or could this have happened by chance?

Design and carry out a simulation to help answer this question.
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STATE

What is the probability that when randomly assigning 12 Math
majors and 6 Stats majors to two teams that there will be one
team with all Math majors?
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PLAN

Using pairs of digits from Table D, we’ll label the 12 math
majors 01–12 and the 6 stats majors 13–18. Number 00 will be
skipped. Moving left to right across a row, we’ll look at pairs of
digits until we have a team of 9.Then we will count how many
different labels from 01–18 we looked at.
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DO


Here is an example of one trial:

Team A: MMMSMMMMM (8 Math majors)

Team B: MSMMSSSSM (4 Math majors)
Since the team with the most Math majors had 8, we will record
the value 8 for this trial. Here are the results of 30 trials:
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CONCLUDE

Since only 1 trial in 30 resulted in a team with all Math majors,
the probability is only approximately 0.033. Since getting a
team of all Math majors is unlikely, we can conclude that the
names were probably not shuffled very well in the hat.
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Golden Ticket Parking Lottery
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 Example:
Read the example on page 290.
What is the probability that a fair lottery would result in two
winners from the AP Statistics class?
Reading across row 139 in Table
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%.
NASCAR Cards and Cereal Boxes
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 Example:
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.
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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