Download Assignment 3: Discrete Probability

Math 340: Discrete Structures II
Due: Tuesday 11th March
Assignment 3: Discrete Probability
1. Bayes Theorem.
(a) Babies. A family has two children, and at least one of them is a boy.
What is the probability that the family has one boy and one girl?
(b) Eye Witness. Consider a large city where 90% of the taxis are black and
10% are blue. One night there is a hit-and-run accident involving a taxi.
An eyewitness identifies the taxi as blue. In tests (under similar conditions)
witnesses correctly distinguish between these two colours 85% of the time.
What is the probability that the taxi was indeed blue?
(c) Drugs Tests. A large company gives a new employee a drug test. The
False-Positive rate is 3% and the False-Negative rate is 2%. In addition,
2% of the population use the drug. The employee tests positive for the
drug. What is the probability the employee uses the drug?
2. The Monty Hall Problem. Consider an alternative version of the problem where
Monty is absent-minded and forgets where the prize is. You select a door.
Monty opens one of the other two doors at random. If Monty’s door contains
the prize you lose and the game is over. On the other hand, suppose Monty’s
door did not contain the prize. Should you now switch doors when offered the
chance?
3. Two Useful Inequalities.
(a) For a finite collection A1 , A2 , . . . , An of events, prove that
P (∪ni=1 Ai ) ≤
n
X
P (Ai )
i=1
(b) Let X be a discrete random variable that only takes on non-negative values.
Prove that
E(X)
∀t > 0
P (X ≥ t) ≤
t
1
4. Geometric Distribution. Suppose we run repeated independent Bernoulli trials
(with success probability p) until we obtain a success. Let the random variable
X be the number of trials needed before we obtain a success.
(a) Calculate the probability P (X = k) for some integer k. [We say that X
has a geometric distribution.]
(b) Prove that E(X) = p1 .
5. Tossing Die. I repeatedly toss a dice until a six occurs for the first time. You
then repeat the experiment. What is the probability that you made more tosses
than me?
6. Balls and Bins. Suppose we have n boxes and we start randomly and independently throwing balls into the boxes.
(a) For a given box, what is the expected number of balls we need to throw
before one of the balls lands in that box?
(b) What is the expected number of balls we need to throw before every bin
contains at least one ball?
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