Transcript Randomness and Probability

Randomness, Probability, and
Simulation
Section 5.1
Reference Text:
The Practice of Statistics, Fourth Edition.
Starnes, Yates, Moore
Activity!
• 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?
Activity!
• 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. Idea of probability
– Law of large numbers
- Probability
2. Myths about Randomness
– Runs “hot hand”
-“law of averages”
3. Simulations
– Preforming simulations
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State
Plan
Do
Conclude
Idea of Probability
• The big fact emerges 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 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.
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%
• 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, and
psychologists tell us that we deal with it
poorly…
Myths About Randomness
• Unfortunately, it is out 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.
Myth: Runs
• The outcome TTHHH in tossing six coins looks
unusual because of the runs of 3 straight tails
and 3 straight heads.
• Runs seem “not random” to out 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 Preforming
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 come
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.
Example: 6 in 1 game!
State
What’s the probability that three of seven people who buy
a 20-ounce bottle of soda win a prize if each bottle has a
1/6 chance of saying, “you’re a winner!”?
Plan
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)
Do
Each student perform 5 repetitions
Conclude
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.
Objectives
• Idea of probability
– Law of large numbers
- Probability
• Myths about Randomness
– Runs “hot hand”
-“law of averages”
• Simulations
– Preforming simulations
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•
State
Plan
Do
Conclude
Homework
Worksheet