University of Illinois ECE 313
at Urbana-Champaign Fall 2007
Problem Set # 4R
Assigned: Wednesday, September 12
Due: Wednesday, September 19
Coverage: This homework set is based on material covered in Chp 4 of Ross (except Sect 4.7),
and pp. 430-431. You are also encouraged to read Chp 8, Sects 8.1-8.3 (pp. 261-269)
of Yates & Goodman, on reserve in the Engineering Library.
In addition to the seven problems below (to be turned in), you are encouraged to attempt problems
in the Self-Test Problems and Exercises section of Chapter 4 of Ross (pages 201-204), particularly
8, 10, 18, 19, 20 and 21; their solutions can be found in Appendix B of Ross.
PROBLEMS
21. In some military courts, 9 judges are appointed. However, both the prosecution and the
defense attorneys are entitled to a peremptory challenge of any judge, in which case that
judge is removed from the case and is not replaced. A defendant is declared guilty if the
majority of judges cast votes of guilty, and s/he is declared innocent otherwise. Suppose
that when the defendant is, in fact, guilty, each judge will (independently) vote guilty with
probability 0.7, whereas when the defendant is, in fact, innocent, this probability drops to 0.3.
(a) What is the probability that a guilty defendant is declared guilty when there are
(i) 9, (ii) 8, and (iii) 7 judges?
(b) If the prosecution attorney does not exercise the right to a peremptory challenge of a
judge and if the defense is limited to at most two such challenges, how many challenges should
the defense attorney make if s/he is 60 percent certain that the client is guilty?
22. A purchaser of transistors buys them in lots of 20. It is his policy to randomly inspect 4
components from a lot and to accept the lot only if all 4 are non-defective. If each component
in a lot is, independently, defective with probability 0.1, what proportion of lots is rejected?
How would the answer above change if the lots are instead of size 100, and the purchaser
adopts still the same policy.
23. Seven people hold reservations for travel in a 4-passenger shuttle from Champaign to O’Hare.
Assume that the number of passengers who actually show up to travel is modeled as a Binomial
random variable with parameters (7,1
2). If more than 4 people show up, only the first 4 get
to go, and the rest are left behind (assume that no two people show up at exactly the same
time). What is the average number of passengers that are left behind?
24. In an experiment, a biased coin with probability of Heads equal to pis tossed until we
observe Heads for the second time. Let Ndenote the number of tosses needed to complete
the experiment. As an example, if the outcomes are T-T-H-T-T-T-H, then N=7.
