Download CS 195 Probability Models and Inference Spring 2013 Homework

CS 195
Homework #2
Probability Models and Inference
Spring 2013
Due: Jan. 30, 2013
The following exercises assume that probability spaces are represented by the notation .; E; P /, where  denotes
the sample space, E the set of events (i.e., subsets of ), and P the probability measure.
1. Two dice are rolled onto a game table. The first die is tetrahedral, consisting of four faces numbered 0, 1, 2,
and 3. The second die is a standard cube, with faces numbered 1 through 6. Thus, any roll can be represented
as an ordered pair .i; j /, where i 2 f0; 1; 2; 3g, and j 2 f1; 2; 3; 4; 5; 6g. Assume the experiment consists of a
single roll of both dice, and that each outcome is equally likely.
(a) Describe the (complete) sample space . What is the value of jj (the cardinality of )?
(b) Let E denote the maximum set of events: E D fA W A g. What is its the value of jEj?
(c) What is the probability that the numbers on the two dice are equal (i.e., “doubles”, if you will)?
(d) Let Ek 2 E, for k 2 f1; 2; : : : ; 9g denote the event that numbers on the two dice sum to k. Thus,
E1 D f.0; 1/g;
E2 D f.0; 2/; .1; 1/g;
E3 D f.0; 3/; .1; 2/; .2; 1/g;
::
:
E9 D f.3; 6/g:
Derive a concise mathematical formula for P .Ek /.
(e) Using your previous result, verify that
9
X
P .Ek / D 1:
kD1
2. In a certain class, 40% of the students regularly take the shuttle bus to class, 80% regularly turn their homework
in on time, and 30% fall into both categories. Find the probability that a randomly selected student neither takes
the bus, nor regularly turns homework in on time.
3. Given two arbitrary events A; B 2 E, show that
P .A [ B/ D P .A/ C P .B/
P .A \ B/:
(If you look carefully, you might find the solution to this problem in the textbook. However, you should express
your solution in your own words and formulas.)
4. Using the previous result, show for A; B; C 2 E, that
P .A [ B [ C / D P .A/ C P .B/ C P .C /
P .A \ B/
P .A \ C /
P .B \ C / C P .A \ B \ C /:
5. Use mathematical induction to prove Boole’s inequality: If Ai 2 E for i 2 f1; 2; : : : ; ng, then
!
n
n
[
X
P
Ai P .Ai /:
i D1
iD1
6. A web service requires each of its 1,000,000 users to create an account with a public username, and a private
password consisting of exactly eight characters chosen from a 64 character alphabet: fa z; A Z; 0 9; ?; !g.
For example, three valid passwords are f94G!foI, ?fhjS3kW, and icecream.
(a) Let  denote the sample space of possible passwords by a particular user. What is the value of jj?
January 22, 2013 (2:46 PM)
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Robert R. Snapp © 2012
(b) There are exactly 29,766 different eight-letter words in an English dictionary, ranging from aardvark to
zyzzyvas. If a password is selected uniformly at random from , what is the probability that it appears in
this dictionary?
(c) As it turns out, most people don’t select passwords at random. A recent study suggests that the three most
common eight character passwords are password (0.20%), sunshine (0.05%), and princess (0.05%).
This vulnerability is commonly exploited by malicious hackers. In order to protect its users, the web server
will lock any account after three incorrect login attempts involving its username. Does this measure provide
effective security? Is there a scheme that would enable a malicious hacker to access a significant number of
user accounts? Explain.
7. An old carnival game requires a table on which is drawn a regular Cartesian grid of 1 inch squares. A player
tosses a dime (diameter 3=4 inch) onto the table. If the dime falls completely within any single square in
the grid, without overlapping any of the lines, then the player wins a prize. Assuming that the dime lands on the
table, estimate the probability that it’s a winner.
January 22, 2013 (2:46 PM)
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Robert R. Snapp © 2012