Math 361 X1
Homework 5 Solutions
Spring 2003
Graded problems:
1;3(a);5(a);6; each worth 3 pts., maximal score is 12 pts.
Problem 1.
Each of the 20 senior professors in a mathematics department is asked to select two (diFerent)
courses out of a total of 100 possible courses that he or she would like to teach in the following
semester. Assume that the professors’ preferences are independent and randomly distributed over
the 100 courses. ±ind the probability that none of the courses has been chosen by more than one
professor.
Solution.
Label the courses 1
,
2
,...,
100, and let (
c
1
,c
2
) be the courses chosen by Professor 1, (
c
3
,c
4
) those
chosen by Professor 2, and so on, with (
c
39
,c
40
) representing the course choice of Professor 20.
Note that, by hypothesis, each of these pairs must consist of distinct numbers, each between
1 and 100 (inclusive). With this notation, a suitable outcome space Ω is the set of all 40-tuples
(
c
1
,c
2
,...,c
39
,c
40
), where the
c
i
’s are integers between 1 and 100, with the restriction that
c
2
6
=
c
1
,
c
4
6
=
c
3
, etc. The number of such tuples is #(Ω) = (100
·
99)
20
(use the slot method, counting
the number of possibilitites for
c
1
,
c
2
, etc.). The event, say
A
, we are interested in corresponds to
the set of those tuples in which
all
c
i
’s are distinct
, so we have #(
A
) =
40
P
100
= 100
·
99
···
61.
Hence the probability that no course has been chosen by more than one professor is
P
(
A
) =
100
·
99
···
61
(100
·
99)
20
= 0
.
000137
...
Remark:
There are several ways to go wrong here. One is not to take into account the requirement
that each professor must choose two
diferent
courses. Without that restriction, #(Ω) would be
100
40
, and the problem would be equivalent to the birthday problem. Another error is to count
#(
A
) using the formula for unordered samples,
(
100
40
)
, with #(Ω) is computed as above, giving
a result that is oF by a factor 40! This violates a cardinal rule when computing combinatorial
probabilities, namely that #(Ω) and #(
A
) must be computed by the same counting technique; if
order is taking into account in #(Ω), then it must also be taken into account in #(
A
).
Problem 2.
A committee of 10 U.S. Senators is selected at random. ±ind the probabilities that the committee
contains (a)
exactly one
, (b)
at least one
Senator from each of the 3 states Illinois, Indiana, and
Michigan. (Note that there are 100 Senators, 2 from each of the 50 states. Part (a) is routine, but