From Algorithms to Z-Scores - matloff - University of California, Davis

2.13. COMBINATORICS-BASED PROBABILITY COMPUTATION 31
3. Consider the example of association rules in Section 2.13.3. How many two-antecedent, twoconsequent
rules are possible from 20 items? Express your answer in terms of combinatorial (“n
choose k”) symbols.
4. Suppose 20% of all C++ programs have at least one major bug. Out of five programs, what is
the probability that exactly two of them have a major bug?
5. Assume the ALOHA network model as in Section 2.1, i.e. m = 2 and X0 = 2, but with general
values for p and q. Find the probability that a new message is created during epoch 2.
6. You bought three tickets in a lottery, for which 60 tickets were sold in all. There will be five
prizes given. Find the probability that you win at least one prize, and the probability that you win
exactly one prize.
7. Two five-person committees are to be formed from your group of 20 people. In order to foster
communication, we set a requirement that the two committees have the same chair but no other
overlap. Find the probability that you and your friend are both chosen for some committee.
8. Consider a device that lasts either one, two or three months, with probabilities 0.1, 0.7 and 0.2,
respectively. We carry one spare. Find the probability that we have some device still working just
before four months have elapsed.
9. A building has six floors, and is served by two freight elevators, named Mike and Ike. The
destination floor of any order of freight is equally likely to be any of floors 2 through 6. Once an
elevator reaches any of these floors, it stays there until summoned. When an order arrives to the
building, whichever elevator is currently closer to floor 1 will be summoned, with elevator Ike being
the one summoned in the case in which they are both on the same floor.
Find the probability that after the summons, elevator Mike is on floor 3. Assume that only one
order of freight can fit in an elevator at a time. Also, suppose the average time between arrivals of
freight to the building is much larger than the time for an elevator to travel between the bottom
and top floors; this assumption allows us to neglect travel time.
10. Without resorting to using the fact that n
k
n
=
k
= n!/[k!(n − k!)], find c and d such that
n − 1
k
+
c
d
(2.66)
11. Consider the ALOHA example from the text, for general p and q, and suppose that X0 = 0,
i.e. there are no active nodes at the beginning of our observation period. Find P (X1 = 0).
12. Consider a three-sided die, as opposed to the standard six-sided type. The die is cylindershaped,
and gives equal probabilities to one, two and three dots. The game is to keep rolling the
die until we get a total of at least 3. Let N denote the number of times we roll the die. For example,
if we get a 3 on the first roll, N = 1. If we get a 2 on the first roll, then N will be 2 no matter what