Problems: Drawing Tickets from a Box Part
2
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13. An insurance company selling term life insurance
finds that one in 99 people who buy the policy will die. The
company must then pay out $20,000. It sells the insurance
policies for $300. If we put this situation into the box
model that the book develops,
 a) How many tickets should be in the box?
 b) How many of those tickets have a $300 on
them?
 c) You might think that there is at least one ticket
with a $20000 on it, but there is not. What is the
correct amount and how many tickets have that
amount?
14. Suppose you are gambling on a coin flip. When a head
comes up, you win a dollar and when a tail comes up, you
lose a dollar. You play this game 100 times.
 a) Playing this game 100 times is like taking 100
draws from the box _______
 b) If you play the game 100 times, the most you could
lose is ________ and the most you could gain is
_________.
 You would expect to break even if you played this
game 100 times, but by random chance you might end up
winning some or losing some. It turns out that we can
estimate quite accurately your chances of winning various
amounts using a normal curve with an average of zero and
a standard deviation of 10. (This week we will see
why!)
 c) If you play the game 100 times, approximately what
is the probability that you will lose $15 or more?
 d) (A working backwards problem) You have a one
percent chance of winning ________ or more.
15. A box has five tickets, 1, 2, 3, 4, 5.
 a. What is the average of the box?
 b) What is the standard deviation of the box?
 c) If 200 draws are made from the box, the sum should
be __________ give or take ____________. (Find the
expected value and standard error of the sum.)
 d) If 200 draws are made from the box, the average of
the draws should be __________ give or take ____________.
(Find the expected value and standard error of the
average.)
 e) If 800 draws are made from the box, the sum should
be __________ give or take ____________. (Find the
expected value and standard error of the sum.)
 f) If 800 draws are made from the box, the average of
the draws should be __________ give or take ____________.
(Find the expected value and standard error of the
average.)
16. Joe and Ted are playing a game with two coins. If
both coins come up heads, Joe pays Ted $1.00. If both coins
come up tails, Ted pays Joe $1.00. If one coin is heads and
one is tails, no payment is made. They play this game 98
times.
 For Ted, playing this game is like taking _______
draws from the box __________. (Be careful.)
Ted's net gain from playing this game 98 times will be
around _________ give or take _________ or so.
17. You are a casino offering a game with ten cards, the
ace through the ten of hearts. The cards are shuffled and
then a player randomly picks a card. If it is the five or
hearts, you (the casino) pay out $80. If it is any other
card, you collect ten dollars. Explaining clearly how you
get your answer, what is the probability that you will be
losing money after 100 plays? After 400 plays? How many
times would you have to play this game so that you have less
than a 1% chance of losing money?
18. In question 1 we looked at the box containing these
five tickets: 0,0,0,0,5. We found the mean was 1 and the
standard deviation was 2. If we draw from this box 400
times, we will expect to get a total of 400 give or take
about 40. If we could repeat this experiment a great many
times, we would end up with a normal distribution with a
mean of 400 and a standard deviation of 40.
 If we draw 400 times from this box, we could get a
total as low as zero and a total as high as 2000. What is
the probability that our draw of 400 tickets from this
box will total more than 500?
19. Chance Pickens pulls a card from a deck of cards. If
it is an A, J, Q, or K, he writes a one on a piece of paper.
If it any of the other nine cards, he writes a zero. He then
shuffles the deck. If he repeats this process 100 times, the
expected percentage of ones in his 100 entries is ________
percent give or take ________ percent. (You will need the
shortcut formula for finding the standard deviation of a
zeroone box. Let p = fraction of ones, and q = fraction of
zeros. The standard deviation is the square root of
p*q.)
With the mean and standard error you have just found, use
the normal curve to approximate the probability that Chance
will get more than 35% successes (writing down a one rather
than a zero) next time he performs his strange exercise?
20. A dice game has the following rules. A single die is
rolled and if a 1, 2 or 3 is rolled, you lose a dollar. If a
4, 5 or 6 is rolled, you win a dollar. If you play this game
150 times, you would expect to win $__________ give or take
$___________.
 a) What is the box?
 b) What is the average of the box?
 c) What is the standard deviation of the box?
 d) What is the expected sum?
 e) What is the standard error of the sum?
 f. Your friend Joe claims that after he played this
game 150 times, he came out ahead by $39 dollars. What is
the probability of getting $39 or more from this
game?
21. A dice game has the following rules. A single die is
rolled and if a 1 is rolled, you lose a dollar. If a 2, 3, 4
or 5 is rolled, you neither lose nor win money. If a 6 is
rolled, you win a dollar. If you play this game 150 times,
you would expect to win $__________ give or take
$___________.
 a) What is the box?
 b) What is the average of the box?
 c) What is the standard deviation of the box?
 d) What is the expected sum?
 e) What is the standard error of the sum?
 f. Your friend Joe claims that after he played this
game 150 times, he came out ahead by $39 dollars. What is
the probability of getting $39 or more from this
game?
22. A dice game has the following rules. A single die is
rolled and if a 1 or 2 is rolled, you lose a dollar. If a 3
or 4 is rolled, you neither lose nor win money. If a 5 or 6
is rolled, you win a dollar. `
 a) How would you represent this game in terms of a
box of tickets?
 b) What is the average of the box?
 c) What is the standard deviation of the box?
 d) What is the expect mean?
 d) If you play the game 150 times, what is the
standard error of the sum?
 e) If you play this game 150 times, you would expect
to win $__________ give or take $___________. (Find the
expected sum for the first blank and the standard error
of the sum for the second.)
 f) Your friend Joe claims that after he played this
game 150 times, he came out ahead by $39 dollars. Do you
consider his claim believable or not? Explain.
23. A person spins a wheel that has an equal chance of
landing on any one of five colors. If it lands on white the
person wins $1, on red $3, on blue $4, on green $5, and on
yellow $7. (Note, this is like picking a ticket from a box
with these five tickets: 1, 3, 4, 5, 7.)
 a) What is the average of the box?
 b) What is the standard deviation of the box?
 c) If the person spins the wheel 100 times, he would
expect to win $____________ give or take about
$_____________. (In the first blank you need the expected
value of the sum of 100 draws from the box, and in the
second the standard deviation of the distribution we get
from drawing 100 times from the box.)
 d) What is the probability that this player will win
more than $450 in those 100 plays?
24. A box with a million tickets has an average of 76
with a standard deviation of 10. If a sample of 400 draws is
made from this box, we expect an average of ______________
give or take ____________or so. The probability that the
average would be more than 77 is
________________________.
25. A box has an average of 12 with a standard deviation
of 3. One hundred draws are made from the box with
replacement and the total is 1173.
This total represents the total of an expected sum of
_______ plus a chance error of _____________. The zscore of
the total is ____________.
26. Suppose that there is a risky investment that has a
90% chance of giving you a 20% return but 10% of the time it
will lose all the money you invested. Imagine investing $100
in this investment.
 a) If we convert this to the box model, what tickets
would be in the box?
 b) What is the average of the box?
 c) What is the standard deviation of the box?
 d) Suppose that we put $100 into two of these
investments. The possible outcomes are lose 200, lose 80,
and gain 40. What are the probabilities for each of these
outcomes? What is the average outcome?
 e) Suppose we have 100 of these investments that are
independent of one another. (This means that if one is a
loss, it does not affect the probability that the others
will be losses.) If we invest $100 in each of these
investments, we would expect to gain ______ give or take
about _______.
 f) What is the probability that you would lose money
with these 100 investments?
 g) Suppose we had 400 of these investments. If we
invest $100 in each of these investments, we would expect
to gain ______ give or take about ________.
 h) What is the probability that you would lose money
with these 400 investments?
 i) In the real world it is dangerous to make the
assumption that investments are completely independent of
one another. Why?
27. Suppose that the show Deal or No Deal has a
contestant who has the following three amounts still on the
board: $75, $50,000, and $400,000. If the offer from the
banker is $130,000, is the offer above or below the fair
offer? By how much?
28. In the game Deal or No Deal, the contestant begins by
choosing one case from 26 cases with the following
amounts:
$.01, 1, 5, 10, 25, 50, 75, 100, 200, 300, 400, 500, 750,
1000, 5000, 10000, 25000, 50000, 75000, 100000, 200000,
300000, 400000, 500000, 750000, 1000000.
 a) Suppose that you want to illustrate this
distribution with a histogram with five classes. What
would it look like? (I am not sure how you can do this
online, other than describing the classes and telling
how many are in each class.)
 b) What is the median amount of this
distribution?
 c) What is the fair value (the mean) of this
game?
 d) Suppose that the contestant opens these six boxes:
$1, $50, $100, $750, $50,000, and $200000. Suppose that
the banker makes an offer of $46000. How much is this
below the fair offer?
29. Suppose a slot machine that cost $1.00 to play will
on average pays out $95. This means that the expected value
or average of each play is $.05. Suppose further that the
standard deviation of each play is $1.50.
 a) If a person plays this machine 100 times, she
would expect to lose about ___ give or take ___.
 b) If a person plays this machine 400 times, we would
expect to lose about ____ give or take ____.
 c) What is the probability that a person would come
out winning money if she played the machine 400
times?
30. A spinning wheel has five colors, each equally likely
to come up. If the wheel stops on red, the player loses
$3.00. If it stops on yellow, the player loses $1.00. If it
stops on black, the player neither loses nor gains. If it
stops on green, the player wins $1.00, and it if stops on
blue, the player wins $3.00.
 a) If we put this situation into the box model, what
is the box?
 b) What are the average of the box and the standard
deviation of the box?
 c) If the player plays 144 times, the expected
average amount he will win is $____ give or take
$_____.
 d) Using the normal curve, approximately what is the
probability that a player playing this game 144 times
will average winnings more than 33 cents?
31. A box has these eight tickets: 1, 1, 1, 3, 3, 3, 5,
7. The average of the tickets is 3. What is the standard
deviation of this set of tickets?
************
32. Today in class we played a game based on DealNo
Deal. We had ten
envelopes with these amounts:
1
2
3
4
5
10
25
50
100
200
Bart selected an envelope. Then he opened three of the
remaining envelopes, which were 1, 5, and 25. The question
was then what was the fair value of the game, or the fair
offer to stop the game. Ten was the median, or middle value,
but this understates the value because the high values are
so large relative to the small ones. To get the fair value,
we found the average or mean of the remaining values, or 369
divided by 7 or about 53. The banker, however, always begins
with an offer much less than the fair value, so we offered
20.
The player then opened two more envelopes, 3 and 100. The
new fair value was then 266 divided by 5, or still about 53.
Since the banker offer usually gets closer to fair value, we
increased the offer to 25.
Eventually the player ended with just two possibilities,
2 and 200. The fair value of this game is 202 divided by 2
or 101. The player rejected the offer of 95 and was left
with 2.
In addition to the mean and median, we talked about
graphing the possibilities to get a histogram, and then
talked about how to measure the variation or dispersion of
the numbers. The range is a quick and dirty way to talk
about dispersion, but for mathematical reasons the standard
deviation, which is not an intuitive measure, is used.
Suppose we have these three groups of numbers, all with
the same average of 40:
39, 41
0, 80
0, 40, 80.
Which has the most dispersion and which the least?
To compute the standard deviation, subtract the mean from
the numbers, then square the results, then add up these
squared numbers, divide them by the number of numbers we
have, and take the square root.
For the first, the answer is 1, for the second it is 40,
and for the third it is 32.66. Is that what you get?
What do you get for these numbers: 0, 40, 40, 80?
 33. Ninetyeight selfprofessed beer lovers were
asked to identify a particular brand of beer in a blind
taste test. There were 3 samples to choose from and 45 of
them correctly identified the beer. Do these results
differ from what we would expect to get by random chance?
Explain.

