Hints for Homework Assignment 1

General remarks

The goal of these problems is to practice the construction of formal probability models in different situations. Coming up with the right answers is the least important aspect here. As pointed out in the instructions for this assignment, you must solve the problems formally by setting up an appropriate probability model, as we have done in the examples worked out in class or the examples in section 1.1 of the text. The construction of a model involves several steps:

Clearly identify all possible outcomes, using correct mathematical notation and explaining your notation For example, if you use (a1,..,a6) to represent an outcome, say what the ai's stand for, e.g., "a1 = birthday of person # 1".
Define an appropriate outcome space Omega. (This is easy once you have done the previous step and correctly identified the outcomes. Just put set braces around the list of outcomes. Of course, you can combine the two steps into one, as I have done in class by writing down the set Omega and identifying the outcomes in this set.
Identify the relevant event(s) as subset(s) of this outcome space (again, using appropriate mathematical notation).
Compute the probabilities of these events within this model.

Several of the problems are modelled after class examples, so you should review those examples before working on the problems.

Hints and comments on specific problems

Problem 2: The maximum of two numbers is the greater of the two; for example, the maximum of 2 and 3 is 3, the maximum of 2 and 2 is 2.
Problem 3: Here "two" is to be interpreted as "at least two", not "exactly two". An equivalent formulation is this: What is the probability that there is (at least) one course that is picked by more than one professor?
Problem 4: In the case when n is bigger than 6 the answer is obvious (no formal justification required), so focus on the case when n is 6 or less.
Problem 5: First, two clarifications: (1) The first student who hears the rumor gets it from someone outside the dormitory. (2) Include the first student among the five to whom the rumor is told.
Next, this problem is conceptually similar to (but not identical) the lottery problem discussed in class. The key to the problem lies the construction of an appropriate space Omega. To this end, you have to know the "rules of the game" which are laid out by the phrase in italics. These rules determine what is and what is not a legal outcome.
Another hint: Draw a picture representing the propagation of the rumor as follows: consider the students, labelled 1 through 10, as balls in a box (the dorm), and draw arrows between these balls to denote the telling of the rumor from one student to another. The first arrow begins outside the box and ends at one of the numbered balls inside the box. The next arrow goes from this ball to another ball, etc. Ask yourself first what type of arrows are not allowed in this construction, according to the rules of the game (one example being a loop arrow from a ball to itself), then determine the restrictions on the sequence of numbers obtained that this imposes. Note also that the answer to the problem depends only on the first six steps (=arrows) in this path.

Last modified Wed 29 Jan 2003 07:38:14 AM CST