University of Illinois Spring 2007
ECE 413: Problem Set 2 Due: 1/31/07 at the beginning of class
So many choices, so little time . . .
This problem set gives you practice in counting and in the calculation of probabilities using
the axioms of probability. It also provides more drill in calculus via a derivation of the
binomial theorem.
Noncredit Exercises: Do NOT turn these in
Chapter 1: Problems 1-5, 7, 9;
Theoretical Exercises 4, 6, 13; Self-Test Problems 1-15.
Chapter 2: Problems 3, 4, 9, 10,11-14;
Theoretical Exercises 1-3, 6, 10, 11, 16, 19, 20; Self-Test Problems 1-8
1. An ice cream manufacturer makes unflavored ice cream and then creates “specialty
flavors” by blending in one or more of the five essences: vanilla, chocolate, fudge, mint,
and almond into the ice cream. How many specialty flavors can the manufacturer
create? Optional noncredit exercise: identify the manufacturer!
2. An experiment consists of observing the contents of an 8-bit register. We assume that
all 256 byte values are equally likely to be observed.
(a) Let Adenote the event that the least significant bit is a ONE. What is P(A)?
(b) Let Bdenote the event that the register contains 5 ONEs and 3 ZEROes. What
is P(B)?
(c) What is P(A∪B)? What is P(A∩B)? What is the probability that exactly one
of the two events Aand Boccurs, i.e. what is P(A⊕B)?
3. Some years ago, the government of a small island nation in the North Atlantic Ocean
decided to change from a constitutional hereditary monarchy to a constitutional ap-
pointed monarchy. Six applicants Andy, Beth, Chuck, Di, Eddie, and Fergie presented
themselves before the House of Commons Search Committee for interviews for the job.
Subsequently, the Committee forwarded a short list of three names to the whole House
of Commons for its decision.
(a) How many different short lists could the Committee have selected?
(b) Assume that all short lists were equally likely to have been chosen.
i. What is the probability that Beth is on the short list?
ii. What is the probability that Chuck is on the short list?
iii. What is the probability that both Beth and Chuck are on the short list?
iv. What is the probability that Chuck and two women are on the short list?
(Those who wish to pretend that they don’t read the National Enquirer are
reminded that Beth, Di, and Fergie are women).
4. (a) For 0 < k ≤n, compute the k-th derivative of f(x) = (1 + x)nusing the chain
rule, that is, without multiplying out the terms to get a polynomial in xand then
differentiating. (In other words, don’t do d
dx (1 + x)2=d
dx (1 + 2x+x2) = 2 + 2x.)
Use these derivatives to find the first n+1 terms of the MacLaurin series (Taylor
series in the vicinity of 0) for (1 + x)n.
