Download 5.1

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.
The Practice of Statistics, 5
t
h
Edition2
What is randomness?
Pick a number:
1
2
3
4
What did you pick?
Almost 75% of people will pick 3. 20% pick 2 or 4. Only
about 5% choose 1!
Give an example of a false positive:
Give an example of a false negative:
The Idea of Probability
Chance behavior is unpredictable in the _________, but has a regular
and ____________ in the long run.
The
law of large numbers
says that if we observe more and more
________ 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 _________ that describes the proportion of times the
outcome would occur in a very _____ series of repetitions.
The Practice of Statistics, 5
t
h
Edition3
Suppose that 4 friends get together to study at Tim’s house for 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?
http://www.rossmanchance.com/applets/randomBabies/Babies.html
Another way to interpret probability of an outcome is its predicted long-run relative
frequency.
For example, if we do many trials of flipping a fair coin, we would expect to see the
proportion of heads to be about .5.
BUT each trial is completely random and not based on any
previous flip or set of flips.
Horse race simulation: We are using the sum of the numbers on a roll of 2
die to simulate horses moving around a track. You can choose to be horse
# 2, 3, 4, ..., 12.
Which number would you choose? Why?
Myths About Randomness
The idea of probability seems straightforward. However, there are
some myths of chance behavior we must address.
The myth of short-run regularity:
The idea of probability is that randomness is predictable in the
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.
The Practice of Statistics, 5
t
h
Edition4
long
What are some myths about randomness?
Myth: Random events are predictable in the short run.
Myth
Myth: A "hot hand" indicates that a streak is likely to continue.
Myth
Myth: The "Law of Averages" says a streak makes other outcomes more likely.
MythRandom events ARE predictable in the long run.
Truth:
Truth: Coins, dice, cards, etc. have no memories. LLN is long run.
Truth
Imagine you are flipping a coin. Write down the results of 50 imaginary flips (e.g.
HTTHT…):
Use technology to simulate: (Write down the steps and results here)
What is the longest run in each set?
HW page 300 (1, 3, 8, 9, 11, 37, 38)
Dear Abby,
My husband and I just had our 8th child. Another girl,
and I am really one disappointed woman. I suppose i
should thank God she was healthy, but Abby, this one
was supposed to have been a boy. Even the doctor told
me that the law of averages was in our favor 100 to
one."
Abigail Van Buren, 1974
Simulation
The __________ 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.
The Practice of Statistics, 5
t
h
Edition5
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?
The Practice of Statistics, 5
t
h
Edition6
5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23
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.
The Practice of Statistics, 5
t
h
Example: Simulations with technology
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.
The Practice of Statistics, 5
t
h
Edition8
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 them up 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, since I already mixed up the names, I don’t want to have separate them all again. Can I just
draw names 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 8 or
more names to get 4 seniors and 2 juniors.
Answer
What are some common errors when using a table
of random numbers?
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.
The Practice of Statistics, 5
t
h
Edition9