EE5137R 2013/14 Problem Set 1

1. (Problem 2.1, 2.5) Consider the experiment of tossing a coin three times.

(a) What is the sample space ? What is its cardinality (i.e. the number of elementsin the set)?

(b) Define event Ei as exactly i tosses yield heads, i = 0, 1, 2, 3. Write down thefour sets E1, E2, E3, E4.

(c) Assume that all of the outcomes in are equally likely. Find P [A], where A =at least two of the tosses are heads.

(d) If the coin is a fair coin, i.e. equally likely to land heads or tails, find theprobability of the elementary (or simple) events in this experiment.

(e) Using the probability assignment in part (d), find P [A].

2. Consider a circular dartboard with unit area. Assume that a dart that is thrown atthe dartboard is equally likely to land anywhere on it, and that all darts thrown landsomewhere on the dartboard.

(a) What is the sample space ?

(b) Find an appropriate -field of events F , and find the probability mapping P :F [0, 1] which models this problem.

(c) On a Venn diagram with the dartboard as , sketch two events A and B whichare independent. State the geometrical relationship between the areas of A, Band A B.

3. From Axioms II and III, the probability of the impossible event, , is 0. If we knowthat P (A) = 0, does this mean that A = ? Explain with the help of an example.

4. The probability space of a problem is (,F , P ). Let = {1, 2, 3}, F contain allsubsets of , A = {1} and B = {1, 3}. Assume too that P [{1}] = P [{3}] = 1/4 andP [{2}] = 1/2.(a) Find P [A|B].(b) Conditioned on B, we have a new probability space (B,G, Q). Find G and Q().

5. A real number in the range (0, 1] is picked at random, with P ((0, t]) = t for 0 < t 1.(a) If you observed only the integer part of the number, would it reduce your un-

certainty about its value? Explain.

(b) If you observed the first decimal place of the number to be 2, use Bayes theo-rem to derive the new probability law you would associate with the experiment.

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6. Consider some disease and its medical diagnosis test. The following statistics areknown:

For a person with this disease, the test yields a positive result 100 percent ofthe time;

For a person without this disease, the test yields a positive result 100 percentof the time;

Of the total population, 100 percent is infected by this disease.

(a) Re-tracing the steps in our in-class example, find

p = P [Disease present|Test positive]

in terms of , and .

(b) Let = 0.99, = 0.02. Plot p as a function of (using Matlab for instance).

(c) Let = 0.01 and = 0.001. Find the smallest value of that makes p 0.9.

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