University of Illinois Spring 2005
ECE 413: Problem Set 6
Due: Wednesday March 9 at the beginning of class.
Reading: Ross, Chapters 3, 4, 5, and the notes on decision-making on the class web page
This Problem Set contains five problems
1. In Problem 3 of Hour Exam I, we considered a tennis game in which Aand Bwin
points with probabilities pand q= 1 −p. If the score reaches deuce (3-3) (an event
Dof probability 20p3q3), then Aand Bwon the game with (conditional) probabilities
p2/(p2+q2) and q2/(p2+q2) respectively. The game can end without the score ever
reaching deuce with scores of 4-0, 4-1, and 4-2 winning for Aand 0-4, 1-4, and 2-4
winning for B.
(a) What are the probabilities that the scores reach these 6 possible terminal values?
(Hint for those whose drug of choice is MTV and not ESPN: the player who wins
the last point in a game also wins the game. Thus, the probability of Awinning
4-2 is not 6
2p4q2.)
(b) What is the probability that Awins the game? Denote this result as f(p).
(c) A little thought will show, I hope!, that Bwins the game with probability f(1−p).
Explain from first principles why f(p) should have the following properties and
determine if your answer to part (b) satisfies these properties:
f(0) = 0, f (0.5) = 0.5, f(1) = 1. Does this mean that f(p) is a linear function of
p?
(d) Find the first two terms of the Taylor series for f(p) in the neighborhood of p= 0.5
and determine the effect on the probability of Awinning the game if p= 0.5 +
where is a small number.
2. The Senate of a certain country has 100 members consisting of 43 Conservative Repub-
licans, 21 Conservative Democrats, 12 Liberal Republicans, and 24 Liberal Democrats.
Before each vote, the groups caucus separately. Each group decides independently of
the other groups whether to support or oppose the motion. All members of the group
then vote in accordance with the caucus decision. If you believe that this is the way politics works, I have
this beautiful skyscraper on Wacker Drive in Chicago that I am willing to sell to you at a real bargain price .. .
(a) Let A, B, C , and Drespectively denote the events that the four groups vote to
eliminate all income taxes on capital gains. Suppose that the probabilities of these
independent events are P(A) = 0.9, P (B) = 0.6, P (C) = 0.5 and P(D) = 0.2.
What is the probability that the bill passes?
(b) The President vetoes the bill as a budget-breaker. Let E, F , G, and Hrespectively
denote the independent events that the four groups support the motion to override
the veto. If these events have probabilities P(E) = 0.99, P (F) = 0.4, P (G) = 0.6,
and P(H)=0.1, what is the probability that the motion to override the veto
passes ?
Political innocents are reminded that a simple majority (51 or more votes) is required
to pass a bill, and a two-thirds majority (67 or more votes) to override a veto.
