Hints for Homework Assignment 2

General remarks

Problem 1: This problem is of the same type as those on the first HW assignment, and should be done in the same manner at the outcome space level (with equally likely probabilities); that is, you need to

clearly identify all possible outcomes, using correct mathematical notation
define an appropriate outcome space (this is easy, once you have done the previous step)
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

Problems 2 - 4: These are theoretical problems, to be done within the Kolmogorov axiom framework, as the examples worked in class last week. In addition to the basic rules (Kolmogorov axioms, inclusion-exclusion, average rule, etc.), you are free to use any properties and rules derived in class, but you should say which rule you are using in each step. For most of these problems, you should draw a picture to see what is going on. For set-theoretic identities (such as (A union B)^c = A^c intersection B^c), a picture suffices as justification.

If a problem does not specifically say that two events are independent, then they are probably not and you should not assume that they are. (As a general rule, you should never add assumptions to those given in the problem. The problems are formulated such that they can be solved with the given assumptions and data, and no additional information is necessary. By the same token, the problems do not contain redundant data or assumptions. If you get an answer without using all of the given data in a problem, you are probably doing something wrong.)

Problems 5 - 6: These are word problems involving rules involving conditional probabilities such as Bayes' rule, the multiplication rule, or the average rule. (We will do several problems of this type in class on Monday and Wednesday. See also Example 3 in Section 1.5.) As indicated on the problem sheet, you should do these problems rigorously within the framework of conditional probabilities. To that end, you need to

Identify and introduce notation for all relevant events.
Translate the given data and the probability asked for in the problem into probabilities (possibly conditional prob.) involving these events.
Apply appropriate rules (in particular, Bayes' rule, but sometimes you will also need other rules such as the complement rule) to compute the prob. asked within the mathematical framework. This is often just a matter of plugging the given data into an appropriate formula, though sometimes some preliminary work is necessary (e.g., applying the complement rule) before the formula can be applied.
Interpret the answer found within the context of the problem.

The first two steps above, which set up the problem as a mathematical problem, are usually the most difficult part of solving the problem, and it is essential that you do this properly. This means, for instance, to know when a verbal description refers to a conditional probability (usually hinted at by words are "if", "given", but it could also arise in connection with "proportions" of subpopulations), and when it refers to probabilities of intersections. (Both instances occur within these problems!)

Bayes' rule depends on having a partition of Omega into sets B1, B2, etc. Often, but not always, this partition consists simply of a set and its complement.

Specific comments

Problem 1: First, the order of the parts (i) and (ii) is deliberate, since there is no (easy) way to solve (ii) directly without first doing (i). In fact, you can think of (i) as a hint, and a key step, in the solution of (ii). Next, it might be helpful to work the problem first with a specific choice of k, say k=13, and then extrapolate the argument to general values of k. In the case k=13, the relevant event becomes "it takes more than 13 people to find a matching birthday". The key is to reformulate this event in a manner that makes it easy to compute its probability. Note that this event depends only on the birthdays of the first 13 people, so you don't need to consider the birthdays of the 14th, 15th, etc. person (in much the same way as for the event "second head occurs at 5th toss" the outcomes of tosses 6, 7, etc., are irrelevant). I can't say much beyond that without giving away the problem.
Problem 2: These are pretty straightforward applications of probability rules. In order to compute the given probabilities, you need to express the sets in question in terms of A, B, and AB. Draw a picture!
Problem 3 (a), (b): This is similar to the proof (given in class) that if A and B are independent then so are A^c and B; the problem asks you to prove the same for A^c and B^c. (While you have to prove this property for this particular problem, you can use it in the remaining problems.)
Problem 3 (b): In class, a formula for P(A^c|B) was derived; this problem asks you to derive a similar formula with B replaced by its complement. You can use the formula derived in class for this purpose.
Problem 3 (c): The argument here is similar to the proof of the average formula, given in class on Friday (which can also be found in the book).
Problem 6: One key here is to correctly express the event "a transmission error occurs" in terms of the given events R1, T1, etc. To that end, ask yourself under what combination(s) of these events you do get a transmission error. Once you have that settled, the computation of the probability becomes a rather simple matter of applying the mentioned rules. Be sure to distinguish between probabilities of intersections and conditional probabilities. You can tell from the wording which of the two probabilities you are dealing with. Intersections are indicated by words like "and", while conditional probabilities are described using words like "if", "given that", or words indicating a subpopulation such as "among", "out of".

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Last modified Mon 03 Feb 2003 06:08:57 PM CST