18.440, Spring 2006.
B
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 then accidentally you put the cards into the envelopes randomly (all
permutations are equally likely).
(a) What is the probability that all 5 of them receive the right card?
(b) What is the probability that none of them receives 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 all three colors occur in the sample?
3. Suppose that 4% of men and 12 % 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 females as males?
4. A car insurance company offers a certain type of coverage with the following conditions:
In the first year, the premium is $W1 . If they charge a premium of $Wi in year i and there
is no accident in that year, then they charge $(λWi ) the following year, where 0 < λ < 1. If
there is any accident in year i, they charge $(µWi ) the following year, where µ > 1. Suppose
that in each year, independently, a car has an accident with probability p.
(a) What is the expected premium in year n?
(b) What is the limit of the expected premium as n → ∞?
5. Peter plays with a fair die. He wins if the first result is 5 or 6, and he loses if it is 1 or 2.
If the first result is 3 or 4, then Peter is eligible to roll again, but now with different rules:
He rolls the die until it is 6, in which case he wins, or until it is the same number as in the
first time, in which case he looses.
(a) What is the probability that Peter wins?
(b) What is the expected number of steps?
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?
(c)∗ What is the probability that there are exactly k worms on the 10 apples?