Math 20: Discrete Probability
Final Exam Solutions
December 3, 2000
1Consider a Bernoulli trials process with probability pfor success (and
probability q= 1−pfor failure). Let Sndenote the total number of successes
in the first ntrials, and let An=Sn/n denote the average number of successes
in the first ntrials.
(a) Show that E(Sn) = np and E(An) = p.
(b) Show that V(Sn) = npq and V(An) = pq
n.
2In the current presidential election, 100,000,000 people voted, and Gore
came out with about 200,000 more votes. Assume that the voting is a
Bernoulli trials with probability pthat a given voter votes for Gore. If
p=1
2, estimate the probability that Gore’s total would be as high as it is
(i.e. greater than or equal to 50,100,000).
3More voting! A popular politician runs for Congress. If she has never
been elected, then the probability that she will be elected is 1
2(and so the
probability that she remains unelected is 1
2and she can run again next time,
in two years). If she has already been elected (and is currently in office) then
her probabiltiy of being re-elected is 9
10 ; the probability that she loses is 1
10 .
If she loses, then she will never be re-elected again, so she retires.
(a) Show how to think of this as a Markov chain. That is, write down the
states and the transition matrix. Explain why the Markov chain is an
absorbing one.
(b) If this is the first year that she runs for Congress, in how many years
should she expect to retire?
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