University of Illinois Fall 2009
ECE 313: Hour Exam I
Monday October 12, 2009
7:00 p.m. — 8:00 p.m.
100 Noyes Laboratory
1. [15 points] Let A,B, and Cdenote three events defined on a sample space Ω, and suppose
that P(A)=0.6, P (B) = P(C) = 0.3, and P(Bc∩C) = P(A∩Bc∩Cc) = 0.2.
(a) [5 points] Find P(B∩C).
(b) [5 points] Find P(B∩Cc).
(c) [5 points] Find P((A∪B∪C)c).
2. [10 points] Aand Bare events defined on a sample space Ω. Assume P(A), P (B)>0.
Mark each of the two statements below as TRUE or FALSE. No justification is needed.
TRUE FALSE
2 2 If P(A|B) = P(B|A), then P(A) = P(B).
2 2 P(A|B)P(B) + P(Ac|B)P(B) = P(B).
3. [30 points] Especially in this problem, you must provide sufficient explanation to justify your
numerical answers.
A fair coin is tossed repeatedly until a Head occurs. Ndenotes the number of tosses.
(a) [5 points] What is the expected value of N?
(b) [5 points] Find the numerical value of P{N > 5}.
(c) [10 points] Given that the event P{N > 5}occurred, what is the expected value of N?
(d) [10 points] Find the numerical value of E[cos(πN )].
4. [20 points] A fair coin is tossed 10 times.
Calculate the probability that the first 5 tosses are all Tails given that a total of 8 Tails
occurred on the 10 tosses.
5. [25 points] Dilbert has 3 coins in his pocket, 2 of which are fair coins while the third is
a biased coin with P(H) = p6=1
2. The probability that a coin chosen at random from his
pocket will land Tails is 7
12 .
(a) [10 points] What is the value of p?
(b) [15 points] Dilbert picks two coins at random from his pocket, tosses each coin once, and
observes a Head and a Tail. What is the conditional probability that both coins are fair?