Chapter 5
5.1 Probability
Probability is the branch of mathematics that
describes the pattern of chance outcomes
As a special promotion for its 20 oz 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.” The prize is a
free 20 oz bottle of soda, which comes out of the store owner’s profits. Seven
friends each buy one 20 oz bottle at a local convenience store. The store clerk
is surprised when 3 of them win. The store owner is concerned about losing
money from giving away too many free sodas. She wonders if this group of
friends is just lucky or if the company’s 1-in-6 claim is inaccurate.
For now, let’s assume that the company is telling the truth, and that every 20
oz bottle of soda as a 1-in-6 chance of being a winner. We can model the
status of an individual bottle with a six-sided die: let 1 through 5 represent
“Please try again!” and 6 represents “You’re a winner!”.
Roll your die seven times to imitate the process of the seven friends buying
their sodas. How many won a prize?
Add your number of wins to the dotplot on the board.
Do this simulation one more time. How many won a prize?
The Idea of Probability
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.
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.
Probability is long-term
relative frequency
Pretend you are flipping a coin 30 times. Write down
the results you see in your mind.
A run is a repetition of the same result. Plot the
length of the longest run you had.
Now use your calculator to simulate these flips. Be sure
you "seed" your calculator. This will insure that the same
random numbers do not appear on everyone's calculator.
Enter a random number in the calculator such as the
student ID number or telephone number. Then press
[STO->] [MATH] "PRB" "1:rand“.
We want to pick 30 numbers, either 1 (let this represent
tails) and 2 (heads). RandInt(1,2,30)" [ENTER]. What is
the length of your longest run?
Myths About Randomness
The idea of probability seems straightforward. However, there are
several myths of chance behavior we must address.
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.
Thinking about randomness
• You must have a long series of independent trials
– The outcome of one trial must not influence the
outcome of any other
Empirical Probability – the probability of
simulations
Theoretical Probability – the probability of
mathematical formulas
• Computer simulations are very useful because we
need long runs of trials
• Short runs give only rough estimates of a
probability
Simulation
The imitation 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.
• Simulation is an effective tool for finding likelihoods of
complex results once we have a trustworthy model
– Table of random digits
– Graphing calculator
– Computer software
• EX: 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? Design and carry out a simulation to help
answer this question.
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Simulation basics
1.
State
I must have all 5 of those NASCAR cards! We will model
picking boxes of cereal from our random digit table with one
of the 5 NASCAR cards in it.
2. Plan
Probability of picking each card is 1/5 so 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.
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Using the RNT
3. Do
Select a line in the RNT (let’s use line 115)
115 61041 77684 94322 24709 73698 14526
We only want to look at numbers between 1 and 5.
61041 77684 94322 24709 73698 14526
4. Conclude
We had to buy 14 boxes of cereal to get all 5 cards. So our
estimate of the probability that it takes 23 or more boxes to
get a full set is very low (roughly 0). That NASCAR fan should
be surprised about how many boxes she had to buy.
Assigning Digits
Consider a random group of people in which 70%
are employed
0, 1, 2, 3, 4, 5, 6 = employed
7, 8, 9 = not employed
Consider a random group of people in which 73%
are employed
Now you have to use 2 digit
numbers (don’t use them unless
you really have to!)
00, 01, 02, 03,…, 72 = employed
73, 74, 75, …, 99 = not employed
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Suppose a cereal manufacturer puts cards of
famous athletes in boxes of cereal in the hope of
boosting sales. The manufacturer announces that
20% of the boxes contain a Alex Rodriguez card, 30%
contain a card of Michael Phelps and the rest
contain a card of Serena Williams. You must have
all three cards! How many boxes of cereal do you
expect to have to buy in order to get the complete
set?
1. State
I want all 3 of those cards so I want to simulate
picking cereal boxes of these probabilities: A-Rod=
20%, Phelps = 30%, Williams = 50%
2. Plan
Assign digits to represent outcomes
A-Rod = 0, 1 Phelps = 2, 3,4 Williams = 5, 6, 7, 8, 9
4. Do
Use line 150 in RNT to perform simulation
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3. Do
07511 88915 41267 16853 84569 79367
This means the order of cards I get is:
AR, SW, SW, AR, AR, SW, SW, SW, AR, SW, MP
4. Conclude
I had to buy 11 boxes of cereal before I had all 3.