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IE 111 Spring 2007 Final Exam Practice
Question 1.You've probably heard the story that if an infinite number of monkeys each
played with a typewriter, they would eventually type out the complete works of William
Shakespeare.
Let's consider just a small case: one monkey, playing randomly with a
typewriter that has only the 26 letters a through z, along with the space bar.
By
"randomly" we mean that he is equally likely to hit any of the keys.
(a)
What is the probability that he types out "two covariance or not two covariance that is
the question" not including the quote marks, exactly correctly on the first try?
Hint: the
quote has 57 characters (letters or spaces).
(b) Suppose instead that he is trying to type the phrase "a rose by any other name"
which has 24 characters.
The probability that he gets this phrase correct on any one try is
4.4388 * 10
35
.
Assuming he tries once each 5 seconds, and each trial is independent,
how long on average will it take until he gets it right?
Express your answer in years.
(c) Which one of our favorite distributions is most relevant to part (b)?
Question 2. Radioactive particles arrive at a Geiger counter according to a Poisson
process, with an average of 4 arrivals per second.
(a)What is the probability of not seeing any arrivals in 2 seconds?
(b)Suppose that we have not seen any arrivals in the last 10 seconds.
What is the
probability of seeing more than two arrivals in the
next 0.3 seconds?
Question 3. We have collected data from various toll booths to get an idea of how long it
takes a car to pay the toll.
(a)For a certain bridge, we have modeled the time to pay the toll as an Erlang random
variable, and estimated that lambda=0.2 per second, and r =3.
What are the mean
and standard deviation (in seconds) of the time to pay?
(b)Using the above parameters, what is the probability that a car leaves in under 5
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 Spring '07
 Storer

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