Part V: Chance Variability

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Part V Chance Variability Chapters 16 to 18
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Chapter 16 Law of Averages
Toss a coin P(head) . 5 P(tail) . 5 Over
many coin tosses of heads half the coin
tosses chance error

• Law of averages
• NOT in the long run, of Heads of Tails
• of heads is around half the number of tosses
• off due to chance error
• as of tosses increases
• errors in absolute terms increases
• with many tosses, (actual - expected) is large
  in absolute terms
• BUT compared to the of tosses, chance error
  becomes smaller
• this error is small relative to the of tosses
  (a of tosses)

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Chapter 16 Law of Averages
Toss a coin P(head) . 5 P(tail) . 5 Over
many coin tosses of heads half the coin
tosses chance error
As the of tosses increase in absolute value,
of errors go up with many tosses, (actual -
expected) is large
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Chapter 16 Law of Averages
Toss a coin P(head) . 5 P(tail) . 5 Over
many coin tosses of heads half the coin
tosses chance error
As the of tosses increase in relative terms,
relative to the of tosses (as a ) errors
decrease
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Chapter 16 Exercise Set A
1. A machine tosses a coin 1,000 times and has
550 heads Express the chance error in
absolute terms and as a percentage of the
number of tosses Repeat for 1, 000,000 tosses,
and 501, 000 heads
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Chapter 16 Exercise Set A
3. A coin is tossed, and you win 1.00 if there
are a. more than 60 heads Which is better
10 tosses or 100 tosses
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Chapter 16 Exercise Set A
3. A coin is tossed, and you win 1.00 if there
are b. more than 40 heads Which is better
10 tosses or 100 tosses
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Chapter 16 Exercise Set A
3. A coin is tossed, and you win 1.00 if there
are c. Between 40 and 60 heads Which is
better 10 tosses or 100 tosses
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Chapter 16 Exercise Set A
3. A coin is tossed, and you win 1.00 if there
are d. Exactly 50 heads Which is better
10 tosses or 100 tosses
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Chapter 16 Exercise Set A
5. A box contains 20 red and 80 blue cards
1,000 cards are drawn at random with replacement.
Which of the following is true and
why? a. exactly 200 cards are going to be
red b. about 200 will be red give or take a
dozen or so
Repeat drawing without replacement with 50,000
cards in the box
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Chapter 16 Law of Averages Chance Processes
Chance Variability when a trial is
replicated over and over, the number of times the
event of interest occurs will differ from the
number of times it is expected to occur
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Chapter 16 Law of Averages Chance Processes
Chance processes are always open to the influence
of chance error gambling roulette, 21, craps
etc tossing a coin sampling randomly
from populations Process for understanding/evalua
ting chance influence (box model) use the
sampling from box analogy allows us to
analyze the chance variability
Example 2 draws made at random with replacement
from
Random drawing process shake box, draw one
card note it down, replace it shake
box, draw one, note it down
1 2 3 4 5 6
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Chapter 16 Law of Averages The sum of Draws
Example 2 draws made at random with replacement
from
Random drawing process shake box, draw one
card note it down, replace it shake
box, draw one, note it down
1 2 3 4 5 6
Now add together the 2 numbers drawn draw
1 2 draw 2 5 7 OR draw 1 3 draw 2
2 5

• replicate this again and again
• each time drawing the same number of tickets
  (here n 2) and noting the sum
• this sum is subject to chance variability
• Note drawing with replacement from this box
  rolling a fair die

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Chapter 16 Law of Averages The sum of Draws
Example 25 draws at random (with
replacement)
1 2 3 4 5 6
About how big is the sum going to be?
In theory 25

• repeat the 25 draws 10 times, calculating the sum
  each time
• sum of first draw of 25 88
• sum of second draw of 25 84
• sum of third draw of 25 80
• sum of fourth draw of 25 90
• sum of fifth draw of 25 78
• sum of sixth draw of 25 94
• sum of seventh draw of 25 94
• sum of eighth draw of 25 80
• sum of ninth draw of 25 89
• sum of tenth draw of 25 83

Actual range 75 100 Question would this change with
replication P(75 ? Note sum of the draws forms the basis for many
of the statistical procedures we use later
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Chapter 16 Law of Averages Making a Box
Model box model allows us to
analyze chance variability three issues to
consider
Question 1 how many tickets in the box?

• Depends on the question
• of tickets reflect the total of possible
  outcomes

rolling a die 6 possible outcomes 6 tickets
Question 2 what values go into the box?

• Depends on the question
• values reflect the amount won/lost on each
  roll
• roll a die
• win 1.00 if you roll a 6
• win -1.00 for 1 through 5 (lose 1.00)

?
-1 -1 -1 -1 -1 1
Question 3 how many draws? of
draws number of plays/trials
The net gain from 100 rolls of the die is like
the sum of 100 draws made at random from the box
net gain cumulative sum of the win-lose numbers
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Chapter 16 Review
1. There is chance error in the of heads in
coin tosses of heads half the number of
tosses chance error 2. Chance error is large in
absolute terms so as the of tosses
increase, so does the of errors between the
actual of heads and the EXPECTED
number 3. Chance error is small in relative terms
so as the of tosses increase, the of
errors relative to the of heads expected
decreases 4. The law of averages with many
tosses, the of heads is likely to be close to
50, though not exactly 50 5. Law of averages
does not change the chances of coming
events 6. Box models model random selection
processes
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Part V Chance Variability Chapters 17
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Chapter 17 The Expected Value and Standard Error
Toss a coin 100 times we expect 50 heads
suppose we get 57 The EXPECTED value for heads
50 The actual number we get varies from the
expected value by 7 Repeat this process over and
over and each time the actual value differs from
the expected value (50) by an amount similar in
size to the SE
Formulas for the expected value and standard
error depend on the chance process that generate
the numbers - here we will use the concept of the
sum of draws from a box Make 100 draws at random
with replacement from the box
1 1 1 5
How large should the sum of draws from this box
be? Well, how can the draws turn out P(1)
3/4 P(5) 1/4 In 100 draws we expect 75(1)
25(5) a sum of draws of 200 sum of the draws
EXPECTED VALUE
The expected value for the sum of random draws
with replacement ( of draws) x (box
average) (100)(2) 200
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Chapter 17 The Expected Value and Standard Error
Roulette example 18 reds, 18 blacks, 2
greens bet 1.00 on red
P(winning 1.00) 18/38 P(losing 1.00)
20/38
Box average 18 (-20)/38 -0.0526 In 100
plays, expect to lose n(box average)
100(-.0526) -5.26 1000 plays, expect to
lose 52.60
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Chapter 17 The Expected Value and Standard Error
0 2 3 4 6
Suppose we make 25 draws at random with
replacement We expect each number to show up
about 1/5th of the time
The expected value for the sum of random draws
with replacement ( of draws) x (box
average) (25)(3) 75 We know that due to
chance error each number will not appear exactly
1/5th of the time - thus the sum of the draws
will not be exactly 75 Recall sum of draws
expected value chance error chance error can
be () or (-) The standard error tells us how big
the chance error is likely to be
Thus a sum is likely to be around the expected
value it differs from the expected value by a
value similar to the SE
Square root law computes the SE for a sum of
draws made at random with replacement SE of
the sum of draws
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Chapter 17 The Expected Value and Standard Error
SE of the sum of draws
the SD (standard deviation) of the list of
numbers in the box measures the spread among the
numbers in the box large spread
large SD difficult to predict how the draws
will turn out which means that the SE will be
large too The number of draws sum of
2 draws is more variable than the sum of 1
draw sum of 100 draws is still more
variable Each extra draw adds some
variability because we do not know how it will
turn out as the number of draws increases, the
sum is harder to predict, chance errors become
larger, and so does the standard error
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Chapter 17 The Expected Value and Standard Error
SE of the sum of draws
Each extra draw adds some variability because we
do not know how it will turn out as the number
of draws increases, the sum is harder to predict,
chance errors become larger, and so does the
standard error

BUT - the standard error increases
SLOWLY SE increases by a factor
So sum of 100 draws is 10 times more
variable than 1 draw
SD versus SE SD is for a list of
numbers SE is for chance variability
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0 2 3 4 6
Earlier example
Suppose we make 25 draws at random with
replacement The expected value ( of draws) x
(box average) (25)(3) 75 The sum of the
draws will be around 75 but will be off by chance
error Recall sum of draws expected value
chance error chance error can be () or
(-) How big is the chance error likely to be????
The standard error tells us how big the chance
error is likely to be Box average 3 Box SD

Square root law states that 25 draws is more
variable by So the SE for 25 draws 5 x 2
10 sum of 25 draws should be around 75 give or
take 10 or so
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Recall our earlier example 25 draws at random
0 2 3 4 6
Expected value (25)(3) 75 SD of box 2 SE
for 25 draws 10 estimates likely size of
chance error
25 draws at random 100 times SD 2 SE
10 Observed values could range from 0
150 All are between 50 and 100
i.e., -2.5 to 2.5 SE Same idea as SD units and
the normal approximation
Ask the precise question what of the observed
sum values fall in a given range, for example
between 50 and 100
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25 draws at random 100 times
0 2 3 4 6
25 draws at random 100 times SD 2 SE
10 Observed values could range from 0
150 All are between 50 and 100
i.e., -2.5 to 2.5 SE Same idea as SD units and
the normal approximation
Ask the precise question what of the observed
sum values fall in a given range, for example
between 50 and 100
Convert to std scores and use the normal curve
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25 draws at random 100 times
0 2 3 4 6
25 draws at random 100 times SD 2 SE
10 Observed values could range from 0
150 All are between 50 and 100
i.e., -2.5 to 2.5 SE Same idea as SD units and
the normal approximation
Ask the precise question what of the observed
sum values fall in a given range, for example
between 50 and 100
Convert to std scores and use the normal curve
0 50 75 100 150
-2.5 0 2.5
99 probability
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25 draws at random 100 times
0 2 3 4 6
Expected value of 1 draw 75 SD 2 SE 10
Observed values could range from 0 150 Same
idea as SD units and the normal approximation
100 observed values 99 should fall 2.5 to 2.5
SEs 99 of the 100 actually do
50 75 100
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25 draws at random 100 times
0 2 3 4 6
Expected value of 1 draw 75 SD 2 SE 10
Observed values could range from 0 150 Same
idea as SD units and the normal approximation
100 observed values 68 should fall 1 to 1
SEs 73 of the 100 actually do
65 75 85
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25 draws at random 100 times
0 2 3 4 6
Expected value of 1 draw 75 SD 2 SE 10
Observed values could range from 0 150 Same
idea as SD units and the normal approximation
100 observed values 95 should fall 2 to 2
SEs 98 of the 100 actually do
65 75 85