# Probability Problem Set 3

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```					Probability: Problem Set 3
Fall 2009
Instructor: W. D. Gillam
Due Oct. 2, start of class

Instructions. Print your name in the upper right corner of the paper and write “Problem
Set 3” on the ﬁrst line on the left. Skip a few lines. When you ﬁnish this, indicate on
the second line on the left the amount of time you spent on this assignment and rate its
diﬃculty on a scale of 1 − 5 (1 = easy, 5 = hard).
We use the conventions q := 1 − p and E := Ω \ E throughout.

(1) Suppose E, F ⊆ Ω are events with P (E) = 1/2, P (F ) = 1/3, P (E ∪ F ) = 2/3.
Calculate
(a) P (E ∩ F )
(b) P (E|F )
(c) P (F |E)
(d) P (E|F ).
Are E and F independent?
(2) Show that the hypergeometric distribution converges to the binomial distribution
as m, N → ∞ and p := m/N remains constant. For simplicity, you may assume
that N and m are simply scaled by a factor of l, so just prove that, for any
k ∈ {0, . . . , n},
lm      lN − lm
k       n−k              n k n−k
lim                       =       p q   .
l→∞          lN                 k
n
(3) Let X be a binomially distibuted random variable with n trials and success prob-
ability p. Show that
1 − q n − npq n−1
P (X > 1|X ≥ 1) =                         .
1 − qn
(4) Assume there are only two types of drivers: bad drivers and good drivers. It
is known that 7/8 of all drivers are good drivers. The probability that a good
driver has an accident in a year is 1/10 while the probability that a bad driver
has an accident is 1/3. A car insurance company charges \$1000 per year times the
probability of having an accident (for example, they charge a good driver \$100 a
year). The car insurance company is not sure whether a certain new policy holder
is a bad driver or a good driver; they only know that he had an accident last year.
How much should they charge him for car insurance?
(5) Suppose, in the previous problem, that the number of accidents (for both good
and bad drivers) has a Poisson distibution (why is this reasonable?) and that a
certain potential new policy holder has had 3 accidents in the past ten years (but
as before, the car insurance company is not sure whether he is a bad driver or a
good driver). What is the probability that this potential new customer is a bad
driver? How much should the insurance company charge him?
1
2

(6) Suppose that 3% of some large number N of people are infected with a disease.
Doctors want to identify these infected individuals by blood testing (assume the
blood test is perfect, returning positive iﬀ the blood is infected). Blood samples
can be pooled and tested all at once—the resulting pooled sample will yield a
positive result iﬀ at least one sample would yield a positive result. They can test
the N people one at a time, or they can divide them into groups of k people (you
can assume k divides N for simplicity), test the pooled samples, then test each
person in a group where the latter test comes back positive. What is the expected
number of tests they need to perform using this second method? What value of k
minimizes this expected number of tests?
(7) Recall from class that we regarded an event E ⊆ Ω with nonzero probability P (E)
as a probability space with sample space E and density PE := P (E)−1 P |E . Given
a random variable X : Ω → R, we may consider the random variable X|E on the
probability space E. Show that E(X) = P (E)E(X|E ) + P (E)E(X|E ).
(8) Establish the formula mn/N for the expected value of the hypergeometric dis-
tribution with parameters N, m, n (“total balls”, “red balls”, “number of drawn
balls”) by using induction on N and the following observation: imagine drawing
the n balls one at a time, so that a typical outcome is a sequence ω ∈ {R, B}n
(with R for red, B for blue; note there is zero probability of drawing more than m
red balls). Consider the event E that the ﬁrst ball drawn is red and use the previ-
ous exercise.1 Note that X|E is hypergeometric with parameters N − 1, m, n − 1,
so its expected value will be known by induction. X|E on the other hand is not
hypergeometrically distributed on the nose, but it diﬀers from a hypergeometric
distribution with parameters N − 1, m − 1, n − 1 by a linear transformation (whose
eﬀect on expected value is known).
(9) Let N be a positive integer and let N = m1 + m2 + m3 be a partition of N into
three parts. Suppose there are N balls in an urn and that m1 of these are red, m2
are blue, m3 are green. Fix n ∈ {0, . . . , N } and a partition of n into three parts
n = k1 + k2 + k3 with 0 ≤ ki ≤ mi . For a randomly chosen set of n balls from the
urn, what is the probability that exactly k1 of the n chosen balls will be red, k2
will be blue, and k3 will be green? Calculate this probability for N = 1 + 2 + 3,
n = 1 + 1 + 1.

1Recall from class that I suggested this approach sheds light on why the expected value is the same whether
or not the balls are replaced: if a red ball is drawn the probability of drawing a red ball on the next draw
decreases, while if a blue ball is drawn, the probability of drawing a red on the next draw increases.

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