Graphical Models:  Assignment 1
Due January 30, 2017

  1. 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. 
    1. 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 ticket. 
    2. 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?
  1. 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.
  1. 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 identify 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 drivers 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 with probability 0.99.  (This number is called the sensitivityof the test -- it is the probability the test will give a positive result if 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 probability 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 than 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.
    1. Find the joint distribution of (D, T)
    2. Find the marginal distributions of D and T
    3. Find the conditional distribution of (D|T)
    4. Comment on your results.