Graphical Models: Assignment 1
Due January 30, 2017
- You are a contestant on a game show. You are given a
well-mixed bucket which contains tickets of different colors.
There are 300 tickets in the bucket. 100
of the tickets are red. The others are green or blue. You do not know
the proportion of green and blue tickets. You will draw a ticket
and receive a prize based on the color of the ticket you draw. Consider
the following two choice problems.
- You are given the choice between: (A) You win $1000 if you
draw a red ticket; and (B) You win $1000 if you draw a blue
- You are given the choice between: (C) You win $1000 if you
draw a red or green ticket; and (D) You win $1000 if you draw a
blue or green ticket.
This problem is a variant of the famous Ellsburg paradox
. Most people who are given these two problems prefer (A) in
the first situation and (D) in the second situation. Explain why a
person who satisfies the axioms of probability and utility might
prefer (A) and (C) or might prefer (B) and (D), but would not
prefer (A) and (D). Do you think it is reasonable to prefer (A) and (D)? Why or why not?
- The famous Monty Hall problem (after the famous game show host Monty
Hall) is stated as follows:
Suppose you're on
a game show, and you're given the choice of three doors: Behind one
door is a car; behind the others, goats. You pick a door, say No. 1
[but the door is not opened], and the host, who knows what's behind the
doors, opens another door, say No. 3, which has a goat. He then says to
you, "Do you want to pick door No. 2?"
Is it to your advantage to
switch your choice? What would you do if
confronted by this problem, and why? Write a clear explanation of
the rationale for your answer.
- In a certain city, police periodically set up a roadblock and use
the same test to take a blood sample of all drivers, seeking to
people driving under the influence of alcohol. It is estimated
that on a typical night, a base rate of
1 in 1000 drivers approaching the roadblock has too
much alcohol in his or
her blood. Experience suggests that 20% of
under suspicion have in fact consumed too much alcohol. If a
driver's blood alcohol is over the legal limit, the test will give a positive result
probability 0.99. (This number is called the sensitivityof
the test -- it is the probability the test will give a positive result
the condition being tested for is present.) If the driver's blood
is not over the legal limit, the test will give a negative result with
0.999. (This number is called the specificity of the test
-- it is the probability
that the test will not give a positive reason for some reason other
the condition being present.) Let
D be a binary (true/false) random variable indicating whether a driver
is driving under the influence of alcohol, and T be a binary random
variable indicating whether the driver tests positive.
- Find the joint distribution of (D, T)
- Find the marginal distributions of D and T
- Find the conditional distribution of (D|T)
- Comment on your results.