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Congrats!
• We are done with Part II of the Topic Outline
for AP Statistics! (10%-15%) of the AP Test
can be expected to cover topics from chapter
4.
• Coming up… Part III (20%-30%) Anticipating
Patterns: Exploring random phenomena using
probability and simulation
Randomness, Probability, and
Simulation
Section 5.1
Reference Text:
The Practice of Statistics, Fourth Edition.
Starnes, Yates, Moore
Consider This…
• The “1 in 6 wins” Game
• As a special promotion for its 20-ounce bottles of
soda, a soft drink company 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.”
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. Is this
group of friends just lucky, or is the company’s
1-in-6 claim inaccurate?
Consider This…
• For now, lets assume the company is
telling the truth, and that every 20-ounce
bottle of soda it fills has a 1-in-6 chance of
getting a cap that says, “you’re a winner”
• We can model the status of an individual
bottle with a six-sided die:
• Let 1 to 5 represent “please try again!”
• 6 represent “you’re a winner”
Activity!
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. Repeat step 1 four more times. In your five repetitions of
this simulation, how many times did three or more of the
group win a prize?
3. Combine results with your classmates. What percent of the
time did friends come away with three or more prizes, just
by chance?
4. Based on your answer in step 3, does it seem plausible that
the company is telling the truth, but that the seven friends
just got lucky?
Objectives
1. The Idea of Probability
-Law of Large Numbers
-Probability
2. Myths about Randomness
– Runs “hot hand”
– “law of averages”
3. Simulations
– Preforming simulations
• State
• Plan
• Do
• Conclude
Idea of Probability
The big fact emerges that when we look at
random sampling or random assignment
closely: chance behavior is unpredictable in the
short run, but has a regular and predictable
pattern in the long run.
This remarkable fact is the basis of probability!
Law of Large Numbers
• The law of large numbers says that if we observe
more and More and More repetitions of any
chance process, the proportion of times that a
specific outcome occurs, will approach a single value.
• The fact that the proportion of heads in many tosses
eventually closes in on .5 is guaranteed by the law of
large numbers.
– With more and more times the situation is being conducted,
the pattern emerges into what's known as the probability.
Flipping a Coin:
Probability of Heads?
Probability
• The probability of an outcome of a chance process is
a number between 0 and 1 that describes the
proportion of times that outcome would occur in a very
long series of repetitions.
0….……………… (.5)………………….1
0% ………………(50%)……………..100%
Facts:
Outcomes that never occur have a probability of 0
An outcome that happens on every repetition has probability of 1
Check for Understanding
• According to the books of odds, the probability
that a randomly selected U.S. adult usually
eats breakfast is 0.61
– Explain what probability 0.61 means in this setting
– Why doesn’t the probability say that if 100 U.S
adults are chosen at random, exactly 61 of them
usually eat breakfast?
Check for Understanding
• 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 occur once
in a while in a long sequence of trials
d) This outcome will occur more often than not.
Myths About Randomness
The idea of probability seems straightforward. It
answers the question:
“what would happen if we did this many
times?”
• Both the behavior of random phenomena and
the idea of probability are a bit subtle.
– We meet chance behavior constantly in our daily
lives… and psychologists tell us that we deal with
it poorly…
Myths About Randomness
• Unfortunately, it is our intuition about
randomness that tries to mislead us into
predicting the outcome in the short run…
• 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!
• Almost everyone says that HTHTTH is more
probable because TTTHHH does not “look
random.”
• In fact both are equally likely. The coin has no
memory, it doesn’t know what past outcomes
were, and it cant try to create a balanced
sequence.
• Myths about Randomness
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.
Randomness, Probability, and Simulation
The idea of probability seems straightforward.
However, there are several myths of chance
behavior we must address.
Myth: “Runs”
• The outcome TTTHHH in tossing six coins looks
unusual because of the runs of 3 straight tails and 3
straight heads.
• Runs seem “not random” to our intuition but are
quite common.
• More examples:
– “she’s on fire!” (basketball player makes several
shots)
– “Law of Averages” Roulette tables in casinos.
“red is due!” from the display board
Simulations and Performing
Them
• Simulation: The imitation of chance behavior,
based on a model that accurately reflects the
situation is called a simulation
Simulations and Preforming Them
Statistics Problems Demand Consistency!
• 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 variables 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.
Example: 6 in 1 Game!
State
Plan
Do
Conclude
What’s the probability that three of seven people who buy a 20ounce bottle of soda win a prize if each bottle has a 1/6 chance of
saying, “you’re a winner!”?
Use a six-sided die to determine the outcome for each person’s
bottle of soda.
6 = wins a prize
1 to 5 = no prize
Roll the dice seven times, once for each person
Record whether 3 or more people win a prize (yes or no)
Each student perform 5 repetitions
Out of 125 total repetitions of the simulation, there were 15 times
when three or more of the seven people won a prize. So our
estimate of the probability is 15/125, or about 12%
Warning: Phrasing Your
Conclusions
• When students make conclusions, they often
lose credit for suggesting that a claim is
definitely true or that the evidence proves
that a claim is incorrect.
• A better response would be to say that there
is sufficient evidence (or there isn’t sufficient
evidence) to support a particular claim.
We will cover this point is a much larger depth when we get
to inference in the 2nd semester.
Objectives
1. The Idea of Probability
-Law of Large Numbers
-Probability
2. Myths about Randomness
– Runs “hot hand”
– “law of averages”
3. Simulations
– Preforming simulations
• State
• Plan
• Do
• Conclude
Homework
5.1 Homework Worksheet
Start Chapter 5 Reading Guide