Hints for Homework Assignment 5
General Remarks
Be sure to study the examples worked in class and the examples on the
handout on combinatorial probabilites before beginning with these problems,
in particular, the "8 days in a year" problem, and box/ball
type problems.
Also keep in mind the two general principles mentioned in class:
(i) In most cases
it is better and safer to work with ordered samples instead of
unordered samples, and (ii) whatever counting method you choose, be
sure to count # Omega and # A by the same method.
Specific Comments
- Problem 1:
Except for one twist, this is a routine birthday type problem.
- Problem 2:
Part (a) is a routine box/ball type problem. Part (b) can be broken into
8 cases each of which is a standard box/ball type problem. An
alternative, but conceptually more difficult approach (not recommended
unless you know what you are doing) would involve the
complement trick and the inclusion/exclusion formula.
- Problem 3:
This is somewhat similar to the "8 days of the year" problem. The
tickets bought by the 10 people are like the days belonging to the
12 months. Think of the winning
tickets as 3 distinct numbers selected at random
after the tickets have already been distributed among the 10
people - just like the 8 days in the "8 days of the year" problem.
In analogy to the method used for the 8 days problem, work with an
outcome space consisting of 3-tuples of distinct numbers between 1 and
100.
- Problem 4:
This problem has a deceptively simple answer, but the solution requires
a very careful argument, via an appropriate outcome space.
Do the problem first for a concrete value of $k$, say $k=3$, then
extrapolate/generalize from there.
The problem is somewhat similar to Problem 2 of HW 2,
a birthday-type problem asking
for the probability that the first matching birthday occurs at the
k-th person) so study that problem first. If you understand that
birthday problem, you should be able to do this one too (though the two
problems are not exactly identical).
- Problem 5:
Part (a) is just like the "8 days of the year"
problem. Part (b) of Problem 5
is analogous to the variant of the 8 days problem in
which the days were not required to be distinct.
- Problem 6:
A relatively easy variant of the birthday problem, and
similar to a problem discussed in class (involving duplicates and
uniques ...). The best approach
is to proceed in stages, via the slot method.
- Problem 7:
No hints for this one since it's an extracredit problem.
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Last modified Sat 01 Mar 2003 02:14:40 PM CST