Chapter 17: The Expected Value and the Standard Error
Ex. #1: Suppose you draw 100 times from a box with four tickets, three
of which are numbered “1” and one numbered “5”. About how large
should the sum of the draws be?
Method #1:
This is the expected value  the value you’d expect after 100 draws.
Observations:
1. The expected value is not the guaranteed value. There is no
guaranteed value  the actual sum should be about 200, but may not
necessarily equal 200.
2. There is a short-cut for finding the expected value:
Expected value = (number of draws) x (average of box)
Method #2:

Average of box =
Expected value =
=

Ex. #2: You bet 95 times on black in roulette. What is your expected
net gain?
Observation: It is okay if this number does not come out even.
Ex. #3: A game is called fair if the expected value for the net gain is 0.
In other words, on average, players neither win nor lose. How much
should the casino offer if it wanted to make roulette a fair game?
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Standard Error
Ex. #1: Suppose 25 draws are made at random with replacement from a
box that contains five tickets. They are numbered “0”, “2”, “3”, “4”,
“6”. What is the expected value of the sum?
Of course, if we actually draw 25 times and add up the tickets, we
probably will not get exactly 75. Remember:
Observed Value = expected value + chance error
How large is the chance error? It should be roughly the magnitude of
the standard error:
standard error 


number of draws SD of box 
Ex. #2: Suppose we look at the previous example again. What is the
standard error?
Solution: We first find the SD of the box:
Deviations from average =

SD =
3

So the standard error is



Conclusion: The sum of the draws is likely to be around 75, give or
take 10 or so.
Ex. #3: Repeat, if 100 draws are made.
Ex. #4: Repeat, if 2500 draws are made.
Notice that the variability increases with the number of draws – but as
the square root of the increase. This is called the square root law.
Observations:
1. The standard deviation (SD) is used for studying lists.
2. The standard error (SE) is used for chance processes – drawing
repeatedly from a list.
3. Just as before, observed values more than 3 SEs away from average
are rare.
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Ex. #5: Fifty draws are made at random with replacement from a box
that contains five tickets, numbered from “1” to “5”. The sum of the
draws turns out to be 157. Fill in the blanks:
The expected value for the sum is
; the chance error is
, the observed value is
; and the standard error is
5
.
Ex. #6: Tickets are drawn at random with replacement from a box of
numbered tickets. The sum of the 25 draws has expected value of 50
and the SE is 10. If possible, find the expected value and SE for the sum
of 100 draws. (Notes: we don’t know the exact contents of the box.)
Ex. #7: Let’s now return to the original problem: drawing 25 times
from a box that contains five tickets, numbered “0”, “2”, “3”, “4”, and
“6”, and considering the sum. What is the probability that the sum will
be between 60 and 90?
Solution: To find this probability, we 1) convert to standard units; then
2) use the normal curve, as we did in Chapter 5.
Expected value = 75; SE = 10
60


in standard units
90


in standard units
So the probability is the shaded area, or 86.64%. This is the power of
the bell curve – it can be used in many different scenarios.
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Ex. #8: One hundred draws will be made at random with replacement
from a box that contains six tickets, two numbered “1”, three numbered
“2” and one numbered “4”.
(A) What is the smallest possible sum? The largest?
(B) What is the expected value? The standard error?
(C) What is the probability that the sum will be bigger than 230?
Ex. #9: You can draw either 10 or 100 times at random with
replacement from a box that contains two tickets, one numbered “  1 ”
and the other numbered “1”. You win a prize if the sum is 5 or more.
How many times should you draw? (No calculation necessary.)
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Ex. #10: Repeat the above experiment. However, this time you win the
prize when the sum is  5 or less.
Ex. #11: Repeat if you win the prize when the sum is between –5 and 5.
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Ex. #12: Thirty-six jets wait to take off from an airport. The average
taxi and take-off time for each jet is 8.5 minutes, with an SD of 2.5
minutes. What is the probability that the total taxi and take-off time for
the 36 jets is less than 320 minutes?
Short cut for Certain Boxes
As we have seen already, finding the SD of a list can be a fairly
laborious process  especially if there are a lot of numbers. However,
there is a short cut for boxes with only two different types of tickets, in
possibly different quantities.
BIG
BIG
SMALL
SMALL

small 
 big


 number number 
SD = 
fraction with
big number
9
x
fraction with
small number
Ex. #13: Find the SD of the ticket in the box with four tickets, three
numbered “1” and one numbered “5”.
Method #1:

Average =
Deviations from average =
SD 
Method #2:

SD  
-








Ex. #14: A gambler plays roulette 100 times, staking $1 on 19 every
time. The bet pays 35 to 1, and he has 1 chance in 38 to win. What is
the probability that the gambler comes out ahead (that is, makes a
positive net gain)?
Notice that, for a small number of plays, coming out ahead in roulette is
entirely possible.
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Ex. #15: Repeat, for 10,000 times. (Roulette wheels are open all day.)
Ex. #16: A die is rolled 60 times. Fill in the blanks: the total of the rolls
should be around __________________, give or take
_________________________ or so.
What box do we use in this case?
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Ex. #17: A die is rolled 60 times. Fill in the blanks: the number of aces
should be around ________________________, give or take
_________________________ or so.
What box do we use in this case?
Observation: You can use the sum of the draws to count the frequency
of an event occurring if you change the contents of the box to 0s and 1s.
This class of problems is called counting problems.
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Ex. #18: A coin is tossed 100 times.
(A) Find the expected value and the standard error for the number of
heads.
(B) Estimate the chance of getting between 45 and 55 heads.
You might note that this is a lot easier to handle than using the binomial
distribution.
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Ex. #19: Suppose 64 students are chosen at random for the student
body. What is the probability that 15 or fewer of them are males?
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