University Problem Set #4 ECE 313
of Illinois Page 1 of 4 Summer 2003
Assigned: Thursday, July 3
Due: Problems 1-4 Monday, July 7; Problem 5-10 Thursday, July 10
Reading: Ross Chapters 3.1-3.5
Noncredit Exercises: (Do not turn these in) Ross Chapter 3: Problems 1,2,5,10,31,38,44,53,62,72.
1. Ross, Problem 5, p. 120 (6th Edition), or p.123 (5th Edition). In addition to the probability
asked for, find the probability that the second ball is red, and determine if it is smaller or
larger or the same as the probability that the first ball is red.
2. The experiment consists of picking a flight at random from all the United Airlines and
America West flights landing at Chicago, Los Angeles, Phoenix, San Diego, or San
Francisco. Let U and W respectively denote the event that the chosen flight is an United
Airlines or an America West flight, let C, L, X, D, and F respectively denote the event
that the chosen flight is landing at Chicago, Los Angeles, Phoenix, San Diego, or San
Francisco, and let T denote the event that the chosen flight is on time. The conditional
probabilities of on-time arrival are as follows:
P(T|UC) = 0.85, P(T|UL) = 0.92, P(T|UX) = 0.95, P(T|UD) = 0.91, P(T|UF) = 0.83,
P(T|WC) = 0.78, P(T|WL) = 0.88, P(T|WX) = 0.92, P(T|WD) = 0.85 P(T|WF) = 0.73.
(a) Based on this data, which airline would you say has better on-time performance? Does
the answer depend on which airport you are talking about?
(b) Use the fact that {C, L, X, D, F} form a partition of the sample space to show that the
average on-time arrival probability P(T|U) for United flights is given by
P(T|U) = P(T|UC)P(C|U)+P(T|UL)P(L|U)+P(T|UX)P(X|U)+P(T|UD) P(D|U)+P(T|UF)P(F|U)
where P(C|U) is the conditional probability that the flight is landing at Chicago given that
it is a United flight etc. State a similar expression for P(T|W) .
(c) 60% of United Airlines flights land at its hub (snowy Chicago), 15% at each of LA and
San Francisco, and 5% at each of Phoenix and San Diego. 75% of America West flights
land at its hub (sunny Phoenix), 10% at LA, and 5% at each of the other three airports.
Use these numbers in the formula of part (b) and show that P(T|U) < P(T|W), i.e., United
has a worse average on-time performance even though it beats America West at all the
five airports! Write a short explanation of the discrepancy between the per-airport on-
time performance and the overall on-time performance.
3. Each box of Cornies, the breakfast of silver medalists, contains one picture, which is of
Luke Skywalker with probability 2/3 and of Darth Vader with probability 1/3,
independently of which picture is in any other box of Cornies. Little Jimmy Kirk of
Cedar Rapids, Iowa, asks his mother to buy boxes of Cornies until he has at least one
picture of both beings, and his mother agrees to do so.
(a) What is the minimum number of boxes of Cornies that Mrs Kirk must buy?
(b) Let X denote the number of boxes of Cornies Mrs Kirk purchases until such time as
Jimmy has acquired at least one picture of each of the two entities. What is the pmf of X?
Verify that the total probability mass specified by your pmf does equal 1.
(c) What is the conditional pmf of X given that the first box contained a picture of Luke?
What is the conditional pmf of X given that the first box contained a picture of Darth?
(d) Use the theorem of total probability to compute the unconditional pmf of X from the
conditional pmfs found in part (c). Do you get the same answer as in part (b)? Why not?
