Stat 304 Take-Home Quiz #1

Stat 304 Take-Home Quiz #1
September 10, 2004
Dr. John G. Del Greco
Instructions.
Name
• You must work alone on this quiz. If you need hints on any of the problems, please
email me or see me during my office hours.
• You should always put maximum effort into the writing up your answers.
will be graded on the quality of the presentation as well as on correctness.
Answers
• This quiz is due by 1:25pm on Wednesday, September 15.
Undergraduate/Graduate Problems
The following problems must be completed by both undergraduate and graduate students.
1. LetE, F ,andG be three events. Find a set expression (using union, intersection,
and complementation) for each of the following events.
(a) only E occurs
(b) both E and G, but not F occurs
(c) at least one of the events occur
(d) at least two of the events occur
(e) all three events occur
(f) none of the events occur
(g) atmostoneoftheeventsoccur
(h) atmosttwooftheeventsoccur
(i) exactly two of the events occur
2. Find the simplest expression for the following events. A simplest set expression is
deÞned to be the one that minimizes the number of sets and set operations.
(a) (E ∪ F ) ∩ (E ∪ F c )
(b) (E ∪ F ) ∩ (E c ∪ F ) ∩ (E ∪ F c )
(c) (E ∪ F ) ∩ (F ∪ G)
3. Let S = {0, 1, 2,...} be a sample space, and deÞne a probability function on S by
P (j) =(1− α) j α for every j, andforsomeÞxed α, 0

(a) Find the probability that the equation has two distinct real roots.
(b) Find the probability that the equation has only one real root.
(c) Find the probability that the equation has complex roots.
5. A number is chosen at random from the set {1, 2,...,1000}. What is the probability
that the number is divisible by either 3 or 5? (Hint: To Þnd the number of numbers
in {1, 2,...,1000} divisible by k, round down the number 1000/k. Also, note that a
number is divisible by 3 and 5 if and only if it is divisible by 15.)
6. Let E and F be events. Prove that P (E ∩ F ) ≥ P (E)+P (F ) − 1. (Hint: Use the
formula for the probability of a union of two events.)
7. Text, p. 49, #2.
Graduate Problems
The following problems must be completed by graduate students only. Undergraduates who
submit solutions to any of the following problems will receive no extra credit for doing so.
8.ProvethatitisnotpossibletodeÞne a probability function on a countably inÞnite
sample space in which each outcome is equally likely. (Hint: Suppose such a probability
function P exits, and let S = {1, 2,...}. Since each outcome is equally likely,
P (i) =α for some α, 0

12. Let f n denote the number of ways of tossing a fair coin n times so that successive heads
never appear.
(a) Prove that
f n = f n−1 + f n−2
for n ≥ 2 if f 0 =1and f 1 =2. (Hint: Break the analysis up into two cases.
The last toss is a tail, and the last toss is a head.)
(b) Find the probability that in 10 tosses of the coin no successive heads appear.
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