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Chapters 16, 17, and 18
Flipping a coin n times, or
rolling the same die n times, or
spinning a roulette wheel n times, or
drawing a card from a standard deck n
times with replacement, …
 Interested in the accumulation of a
certain quantity? We can box model the
 Actual outcomes are therefore abstractly
represented by tickets.
to Make a Box Model
First draw a rectangular box.
 Then write next to the box how many
times you are drawing from it: n = …
 What tickets go inside the box?
 That depends on what value you could add
on to a requested quantity each time you
draw from the box!
 Then, write the probability of drawing a
particular ticket next to that ticket.
 Examples: Chapter 16, #5-8
The expected value of n draws from the
box is therefore given by:
 EVn = n*EV1
The expected value of 1 draw from the
box, also called the box average, is
given by:
 EV1 = weighted average of tickets in box
 = first ticket *probability of drawing first ticket +
second ticket *probability of drawing second ticket +
“The more you play a box-model-appropriate
game, the more likely you get what you see of
the box.”
 “What is expected to happen will happen.”
 Examples: Chapter 16, #1, #4
 A consequence of the Law of Averages is
that we should not hope to come away with
a gain by playing many times – we will
eventually come out as a loser if we play
long enough.
The expected value of n draws is given
by EVn = n*EV1, where EV1 is the
average of the box.
 Now of course our actual accumulated
total could differ somewhat from the
expectation, and we call our typical
deviation standard error, given by:
 SEn = √n *SE1, where SE1 is the
standard error of the box.
Standard error of a box, or standard error
of a single play, or standard error of a
single draw, all mean the same thing.
 For a box with only two kinds of tickets,
valued at A and B respectively, and with
probability of p and q of being drawn
respectively, the standard error of the box
is given by:
 SE1=|A-B|* √(p*q)
 Examples: Chapter 17 #10
This is related to the normal table we played with.
Now the EVn acts as the “Average”
And SEn acts as the “Standard Deviation”
Chapter 17, Question 3c
Continuity Correction is needed when you are
dealing with discrete outcomes.
Suggestion: Draw the normal curve and label the
average. Then judge where you want to be and in
what direction you should shade; then standardize
and look up percentages.
And so the new version of Standardization Formula:
actual  EV n
SE n
When do we know we may use continuity correction?
That’s when the observed outcomes are discrete.
For example, if you are counting the number of democrats
among a sample of 400 people, you can probably get 0,
1, 2, …, 399 ,or 400, but nothing else between any two
numbers (such as 349.97)
Examples: All questions in Chapter 18 where the box
model is a “COUNTING BOX” (box with only 0 and 1)
A non-example: Height of people in the US
Example: P(at least break even) = P(actual > -0.5)
Example: P(lose more than $10) = P(actual < -10.5)
Example: P(win more than $20) = P(actual >20.5)
Example: P(no more than 2300 heads) = P(actual <
The basis for what we did is called
Central Limit Theorem.
The Central Limit Theorem (CLT) states that
 We play a game repeatedly
 The individual plays are independent
 The probability of winning is the same for each play
Then if we play enough, the distribution for the
total number of times we win is approximately
 Curve is centered on EVn
 Spread measure is SEn
Also holds if we are counting money won
Note: CLT only applies to sums! See Chapter
18 Question 10.