Math 218
Final Exam Solved
Fall 1999
The solutions which we reproduce below are far more elaborate than what
was expected on the final. After all, not many students have laser printers and
Mathematica running on a laptop during the exam! Our purpose is to give as
full an explanation as possible, so
you
understand the solution.
Problem
1
.
(15 points) A certain investor is interested in a group of 30 stocks.
Among these stocks are 17 stocks traded at the New York Stock Exchange
(NYSE) and 13 stocks traded at the American Stock Exchange (Amex): Among
the NYSE stocks, the prices of 12 stocks are up since the previous trading day
and the prices of 5 stocks are down. Among the Amex stocks, the prices of
7 stocks are up since the previous trading day and the prices of 6 stocks are
down. When appropriate, you may leave your answer in terms of factorials.
(a) Suppose one stock is selected at random. Let
A
represent the event that
a stock traded on the NYSE is chosen, and let
B
represent the event
that a stock whose price has risen is chosen. Find
P
(
A
)
and
P
(
A

B
)
.
(b) Are
A
and
B
independent? Show your work.
(c) The investor needs to read a report on each of the Amex stocks whose
price is up since the previous trading day, and must choose the order in
which to read them. How many ways are there for the investor to do
this? Show your work.
(d) How many ways can the investor choose a group of 5 different stocks
to buy from the group of 13 Amex stocks?
(e) Suppose that the investor wants to invest in some of the 13 Amex
stocks. How many ways can the investor choose a group of 3 Amex
stocks that are up and a group of 2 Amex stocks that are down?
Solution.
(a) There are a total of 30 stocks; the event
A
represents choosing one
of the 17 NYSE stocks, so
P
(
A
)
=
17
/
30
=
0
.
5667, the number of
successes over the number of possibilities.
By definition,
P
(
A

B
)
=
P
(
A
and
B
)/
P
(
B
)
, where
B
is the event
“the price has risen”. There are 19 stocks which are up, of which 12
are NYSE stocks; thus
P
(
A
and
B
)
=
12
/
30, and
P
(
B
)
=
19
/
30.
Therefore
P
(
A

B
)
=
12
/
19
=
0
.
6316. Of course, this has also the
commonsensical solution: probabilities
given B
are over a new uni
verse of 19 simple events (the stocks that went up), and 12 of them
were from the NYSE, thus the probability of choosing an NYSE stock
from the 19 which went up is 12
/
19.
(b) For
A
and
B
to be independent, we must have
P
(
A
and
B
)
=
P
(
A
)
P
(
B
)
(because that’s the
definition
of independent events). So we are asking
whether
12
19
=
17
30
×
19
30
,
and this is certainly false. (For one thing, the numerator on the right
side isn’t divisible by 3, and the numerator on the left is. Or pull out
your calculator and check.)
(c) 7 of the Amex stocks are up, and you’re asked how many ways the
investor can order these 7 stocks. The answer is
7!
=
1
×
2
×
3
×
4
×
5
×
6
×
7
=
5040
Perhaps he should quit daydreaming and read them alphabetically
;)
(d) This is
(
13
5
)
(which is even
pronounced
“13 choose 5”), or
µ
13
5
¶
=
13
·
12
·
11
·
10
·
9
1
·
2
·
3
·
4
·
5
=
1287
ways. (You can also use
µ
13
5
¶
=
13!
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 Spring '06
 Haskell
 Math, Calculus

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