Stat 20: Intro to Probability and Statistics
Lecture 15: Law of Averages
Tessa L. Childers-Day
UC Berkeley
21 July 2014
Exams
Today’s Goals
Recap
Law of Averages
Box Models
Exam Performance
In general, scores were about typical
Total Possible Points = 70
Minimum = 31 points
Mean = 50.47 points
Median = 52 points
SD = 9.52 points
Maximum = 67 points
The maximum possible points were achieved for each problem
by at least one student
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Exam Performance (cont.)
0.05
Objective Grades
Using the RAW scores:
38.5% of class scored
below 49 (D or F)
0.03
Density
0.02
0.01
28.9% of class scored
49 to 56 (C)
0.00
23% of class scored 56
to 63 (B)
0.04
9.6% of class scored
63 or above (A)
A (9.6%)
B (23%)
C (28.9%)
30
40
50
60
70
Total Score
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Exam Performance (cont.)
0.05
Adjusted Grades
7.7% of class scored
below 34 (D or F)
Density
0.02
0.01
0.00
38.4% of class scored
34 to 52 (C)
0.03
21% of class scored 60
or above (A)
32.6% of class scored
52 to 60 (B)
A (21%)
B (32.6%)
C (38.4%)
0.04
ROUGHLY speaking, this
translates to:
30
40
50
60
70
Total Score
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Exam Performance (cont.)
If you scored below a 40 OR you want to improve your score, ask:
1
Are you attending lecture and section?
2
Are you paying attention/taking notes in lecture and section?
3
Are you participating/asking questions in lecture and section?
4
Are you reading the book and doing the practice problems
(unassigned exercise sets)?
5
Are you attending office hours to ask questions?
You should be doing ALL of these things to maximize your
understanding (and thus your grade)
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Re-grade Policy
You are welcome to request that your exam be re-graded, but the
following guidelines apply:
Re-grade requests will be accepted only in my office hours
from Tuesday, July 22 to Thursday, July 24
Your request must be in writing, and state why you believe
you deserve more points on each additional question. It must
specifically compare your answers to the answers in the posted
solutions.
Your exam will be re-graded in its entirety. Your overall score
may go up, down, or stay the same.
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
By the end of this lecture...
You will be able to:
Relate the law of averages to the definition of probability
Explain what the law of averages does and does not say
Draw a box model to analyze games of chance
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
The Course So Far
Thus far we have learned how to:
Design experiments and surveys
Display and summarize data
Use one variable to inform about another
Calculate probabilities for games and simple situations
Next step: Calculating more complicated probabilities, using box
models
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Probability Theory
Recall the frequency theory of probability: probability is the
limit of the relative frequency with which an event occurs, in
repeated trials
P(event A) = lim relative frequency of A
n→∞
= lim
n→∞
# of times A occurs
# of trials
As we perform more and more trials, the quantity
# of times A happens
gets closer and closer to P(event A).
# of trials
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
The Law of Averages is...
Simple extension of frequency theory of probability. As the number
of trials increases:
the relative frequency of an event
# of times A happens
# of trials
gets closer and closer to P(A)
the difference between the relative frequency of an event and
the probability of the event gets smaller (closer to 0)
NOT relative frequency of A = probability of A
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
The Law of Averages is NOT...
A statement about equality or exactness. As the number of trials
increases:
the relative frequency of an event DOES NOT equal P(A)
the difference between the relative frequency of an event and
the probability of the event DOES NOT equal 0
the difference between the number of times an event occurs
and the number of times we expect the event to occur DOES
NOT equal (or even get closer and closer) to 0
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Chance Error
Let’s define
chance error = # observed – # expected
If we are rolling a fair die, then P(roll a 6) =
?
If we roll a fair die 60 times, how many “6”s do we expect?
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Example: Dice Throwing
If we roll a fair die many, many times, what does the Law of
Averages tell us?
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Example: Dice Throwing (cont.)
0.22
0.18
0.14
Relative Frequency of "6"s
Relative Frequency
0
2000
4000
6000
8000
10000
0.02
0.06
Relative Frequency − Probability
−0.02
Relative Frequency of "6"s − 1/6
Number of Tosses
0
2000
4000
6000
8000
10000
Number of Tosses
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Example: Dice Throwing (cont.)
If we roll a fair die many, many times, what doesn’t the Law of
Averages tell us?
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Example: Dice Throwing (cont.) (cont.)
500 1000
0
Frequency of "6"s
Frequency
0
2000
4000
6000
8000
10000
Number of Tosses
−10
−30
−50
Frequency of "6"s −
(1/6)Number of Tosses
Frequency − Expected Frequency
0
2000
4000
6000
8000
10000
Number of Tosses
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Example: Dice Throwing (cont.)
1
If you roll a fair die 60 times and observe 12 “6”s, what is
your chance error?
2
You win $10 if 20% or more rolls come up “6”. Would you
rather roll 60 times or 600 times?
3
You win $10 if between 15% and 18% of rolls come up “6”.
Would you rather roll 60 times or 600 times?
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Example: Losing Streaks
You and a friend are flipping coins for money. If it lands “heads”
you win $1 from your friend. If it lands “tails” you give your friend
$1. You have given your friend $10 in 10 coin flips.
You decide to stop, since you’re on such a losing streak. Your
friend says you should keep playing, because you are due to win
soon (from the law of averages).
Which of you is right? Or are you both wrong?
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Drawing Box Models
Recall box models for probability.
Make chance processes into draws from a box
Connect variability of interest to variability of box
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Drawing Box Models (cont.)
1
Draw a box
2
Randomly pull desired number of tickets from box
3
Do something with the tickets
Often used in gambling problems:
net gain = money earned - money spent
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Drawing Box Models (cont.)
Say we are interested the sum of 3 dice rolls. That is, roll the die 3
times, add up the spots on all of the faces.
What does our box look like?
How many tickets do we draw?
How big can the sum be?
How small can the sum be?
Will every draw be the same?
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Examples
For each situation please specify:
The composition of the box
How many tickets we draw
The minimum and maximum possible
1
Rolling a die 10 times, and adding up the spots seen
2
Rolling a die 10 times, and counting up the even spots seen
3
Rolling a die 10 times, winning $5 if you see an even number,
or losing $2 if you see an odd number
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Examples (cont.)
A box contains 500 tickets: 300 “0”s and 200 “1”s. 500 draws will
be made with replacement. Which of these best describes the
situation, and why?
1
The number of “1”s will be 200 exactly
2
The number of “1”s is likely to be 200, but there is a small
chance it will be something else
3
The number of “1s” is likely to be different from 200, but the
difference will probably be small compared to 500
What if we draw without replacement instead of with replacement?
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Examples (cont.)
A box contains 5 tickets: -4, -2, 0, 2, 4. 500 draws will be made at
random with replacement from the box.
1
If the sum of the 500 numbers drawn is 40, what is their
average?
2
If the sum of the 500 numbers drawn is -24, what is their
average?
There are three ways to win $100
3
(a) If the sum of the 500 numbers drawn is between -10 and +10
(b) If the average of the 500 numbers drawn is between -0.02 and
+0.02
(c) If the average of the 500 numbers drawn is between -0.05 and
+0.05
Which of these is best? Or are they all equal? Explain.
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Exams
Today’s Goals
Recap
Law of Averages
Box Models
Important Takeaways
The law of averages: if an experiment is repeated, as the
number of trials increases, the relative frequency of an event
gets closer and closer to the probability of that event
Chance error = # observed - # expected
Box models: indicate the number, and kind of each ticket, the
number and kind of draws, what is done with the ticket
Next time: More about box models.
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