C22.0103
Homework Set 2
Spring 2012
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1. MBS11, problem 3.18, page 129.
2. MBS11, problem 3.21, page 130. Note that the inclusion of grill #2 is required in the
selection.
3. MBS11, problem 3.23, page 130. This problem claims to be “advanced,” but it isn’t.
Here’s a simple restatement. If E is an event with probability P(E), then the odds
on E are given as P(E) : 1 – P(E), expressed in integer terms if possible. For
instance, if P(E) = 0.40, then the odds on E (or the odds in favor of E) are
0.40 : 0.60. This would be more easily expressed as 2 : 3 and read as “2 to 3.” It
could also be said that the odds are “3 to 2 against E.”
If the odds on E are given as a : b, then P(E) =
a
.
ab
4. Suppose that a race involves four horses, and that the probabilities of winning are,
respectively, a, b, c, and d. It’s easy to see if this is consistent just by checking whether
a + b + c + d = 1. The checking is not so easy in terms of odds.
a.
Bettor X gives the odds against the four horses as 11 to 1, 2 to 1, 31 to 5,
and 5 to 4. Are these consistent?
b.
Bettor Y gives the odds against the four horses as 6 to 1, 2 to 1, 8 to 1, and
3 to 1. Are these consistent?
c.
Suppose that bettor Z gives the odds against the first three horses as 3 to 1,
5 to 1, and 2 to 1. What would be his odds against the fourth horse if he is
to be consistent?
5. If one card is selected at random from a standard deck of 52, find the probability
that
a.
the selected card is a diamond () ;
b.
the selected card is a seven;
c.
the selected card is red;
d.
the selected card either a face card or an ace.
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C22.0103
Homework Set 2
Spring 2012
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6. A coin is to be flipped five times, in sequence. What is the probability that the
number of heads will always exceed the number of tails? The sequence HHTHH
satisfies the description, as does HHTHT.
In the sequence H H T H H, the running scores in favor of heads are
1-0, 2-0, 2-1, 3-1, 4-1
and H is ahead throughout.
In the sequence H H T H T, the running scores in favor of heads are
1-0, 2-0, 2-1, 3-1, 3-2
and H is ahead throughout.
However, in each of the sequences HTHHH, TTHHH, and HHTTH there is a point at
which the number of tails equals or exceeds the number of heads.
7. This problem deals with a laboratory situation involving a monkey and three M&M
candies of different colors. Monkeys like M&Ms, as do most children. In fact, a
separate part of the research work used four-year-old children as subjects.
The work is by Louisa Egan, Laurie R. Santos, and Paul Bloom, “The Origins of
Cognitive Dissonance: Evidence from Children and Monkeys,” Psychological
Science, Vol 18, No. 11, pp 978-983, November 2007.
Use letters A, B, and C to refer to the colors. On the first round of the experiment, the
monkey will be given a choice between two of the colors and only allowed to have the
chosen color. In the experiment, this was accomplished by placing the two candies in
separate trays, arranged so that the monkey could only get to one tray. On the next
round, the monkey will be given a choice between the color rejected at the first round and
the novel color.
Suppose that the monkey is given a choice between A and B on the first round and
that the monkey selects A. On the second round, the monkey will be given a
choice between the color rejected at the first round (B) and the novel color (C).
Let  be the event “monkey chooses the novel color on the second round.”
a.
b.
c.
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Suppose that the monkey is completely indifferent to the colors A, B, C.
Find P().
Suppose that the monkey has a strict preference ordering. Find P(). This will
be much easier if you assume that the preference ordering is
A (best) > B > C (worst)
Now suppose that the randomness comes from the experimenter’s selection of the
two colors to present at round one. That is,
with probability 13 the monkey will be asked to choose between A and B
with probability 13 the monkey will be asked to choose between A and C
with probability 13 the monkey will be asked to choose between B and C
Suppose that the monkey has a favorite color, but is indifferent to the other two
colors. Find P(). This will be much easier if you assume that A is the favorite
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d.
e.
Homework Set 2
Spring 2012
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color and that the monkey is indifferent between B and C. Assume also that a
choice between B and C is equivalent to a coin flip.
Suppose that the monkey dislikes one color, but is indifferent to the other two
colors. Find P(). This will be much easier if you assume that C is the least
favorite color and that the monkey is indifferent between A and B. Assume also
that a choice between A and B is equivalent to a coin flip.
Suppose that the monkey’s preferences are circular. Let’s say that
the monkey prefers A over B
the monkey prefers B over C
the monkey prefers C over A
Find P().
In working through these problems, you might find it helpful to set up tables with these
columns:
Round
one
choice
Novel
color to be
used for
round two
Probability
Round
one
monkey
selection
Round
two
choice
Round
two
monkey
selection

occurs?
There is a bit of scientific intrigue to this story. Details will be provided with the
solutions.
8. An janitorial supply firm sometimes gets calls related to improperly filled orders.
This situation is related to the salesperson’s error in writing up the bill of sale. It happens
that Herb will make an error on the bill of sale with probability 0.07, Jack will make an
error on the bill of sale with probability 0.03, and Charlie will make an error on the bill of
sale with probability 0.11.
It also should be noted that Herb writes 30% of all sales, Jack writes 30% of all sales, and
Charlie writes 40% of all sales.
If the firm receives a call about an improperly filled order, what is the probability that the
bill of sale was written by Herb? …by Jack? …by Charlie?
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C22.0103
Homework Set 2
Spring 2012
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9. All four hundred students at a certain junior high school were asked whether the
school principal should be allowed to edit out political content in the school newspaper.
The responses were these:
Grade
7
8
TOTAL
Principal should be
allowed to edit out political
TOTAL
content in student
newspaper
YES
NO
70
140
210
20
170
190
90
310
400
Assume that this listing represents a probability mechanism. That is, selecting any one
student at random exactly matches probabilities. Then find
a.
the probability that a student would say YES;
b.
the probability that a grade 7 student would say YES;
c.
the probability that a student saying YES is in grade 7.
10. In a certain town, 60% of the voters favor candidate A and 40% favor candidate B.
Of those who favor A, it happens 82% support proposal X, which will cause a
reassessment of all the homes in town. Of those who favor B, only 36% support
proposal X. What is the probability that a person who supports proposal X is also in
favor of candidate A?
11. You are presented with two decks of cards. These look identical, but they are not.
One deck contains all 52 cards. The other deck has only 48 cards, as it is missing all
the 7s. You select one of the decks at random.
a.
What is the probability that you have selected the defective deck?
b.
You are allowed to select one card from your chosen deck and turn it over. This
chosen card is a 9. Knowing this, what would you now give for the probability
that you’ve selected the defective deck?
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