18.440, Spring 2006.
A
Exam 1
Instructions. Solve all six problems one by one. Work on these sheets only. Do not use
any books, prepared notes, or calculators. Do not consult your classmates. Hand in your
solutions before 4pm. Each problem is worth 10 points, the maximum total score is 60.
Name:
1.
2.
3.
4.
5.
6.
Σ:
1. You want to send 5 greeting cards to 5 friends. You have written the cards and addressed the envelopes. But, accidentally, you put the cards into the envelopes randomly (all
permutations are equally likely).
(a) What is the probability that at least 3 of your friends receive the right card?
(b) What is the probability that at most 3 of them receive the right card?
2. There are 3 red, 3 blue, and 3 yellow balls in a bin. You draw a random sample of 4 balls.
What is the probability that only two colors occur in the sample?
3. Suppose that 6% of men and 1% of women are colorblind. A colorblind person is chosen
at random. What is the probability of this person being male with each of the following
assumptions?
(a) Assume that there are an equal number of males and females.
(b) What if the population consisted of twice as many males as females?
4. You are searching for a document, which is in drawer A with probability pa and in drawer
B with probability 1 − pa . It takes ta minutes to search drawer A and tb minutes to search
drawer B, independently of whether the document is found or not. You have two strategies:
search drawer A first and then B, or vice verse. Which strategy leads to a shorter expected
search time? Give conditions in terms of pa , ta , and tb for the first or the second strategy
being better, and for both being equally efficient.
p
5. A biased coin, where the odds of heads is 1−p
, is flipped repeatedly. We partition the
sequence of heads and tails into runs, which are maximal subsequences of consecutive heads
or tails. Let X denote the length of the first run, and let Y be the length of the second run.
(For example, the sequence HHT T T HT HHT . . . gives X = 2 and Y = 3.)
(a) Determine E[X].
(b) Determine E[Y ].
6. In an apple farm, the number of a certain type of worms on each apple is a Poisson
random variable with λ = 2. The apples are treated with insecticide that kills each worm
independently at random with probability p = 34 . You buy 10 apples from this farm.
(a) What is the probability that there is no worm on your apples?
(b) What is the probability that there is exactly one worm altogether on the 10 apples?
Bonus problem: What is the probability that there are exactly k worms on the 10 apples?