COU 1: Modeling Uncertainty

In answering this prompt and all the other prompts in this COU, make sure you understand and correctly apply the definition of a probabilistic model of uncertainty and the definition of a probability distribution.

Also, in responding to the prompts in this COU, keep the academic honesty requirements for this course in mind: You may not share an answer that you develop for any COU with another student, and you may not consult an answer developed by another student. There are an infinite number of correct answers to each of Prompts 1A, 1B, 1C, 2A and 2B in this COU. Thus we will suspect any pair of students who submit the exact same answers to any one of those prompts of violating the course’s requirements.

Part 1

Imagine an election in which four candidates are running, named Jamal, Katherine, Lucretia and Marisol.

Prompt 1A

Write a probabilistic model that depicts uncertainty about which of the four candidates will win the election, assuming that exactly one of the candidates will win the election. The model you write should depict a situation in which Jamal is more likely to win the election than Katherine, who in turn is more likely to win the election than Lucretia, who in turn is more likely to win the election than Marisol.

Prompt 1B

Write a probabilistic model that depicts uncertainty about which of the four candidates will win the election, assuming that exactly one of the candidates will win the election. The model you write should depict a situation in which Jamal is less likely to win the election than Katherine, who in turn is less likely to win the election than Lucretia, who in turn is less likely to win the election than Marisol.

Prompt 1C

Write a probabilistic model that depicts uncertainty about which of the four candidates will win the election, assuming that exactly one of the candidates will win the election. The model you write should depict a situation in which Jamal and Katherine are equally likely to win the election, Katherine is less likely than Lucretia to win the election, and Lucretia and Marisol are equally likely to win the election.

Prompt 1D

Note that there is only one correct answer to this prompt

Write a probabilistic model that depicts uncertainty about which of the four candidates will win the election, assuming that exactly one of the candidates will win the election. The model you write should depict a situation in which Jamal is twice as likely to win the election as Katherine, Katherine is twice as likely to win the election as Lucretia, and Lucretia is twice as likely to win the election as Marisol.

Part 2

Prompt 2A

In Lesson 3, we studied a model in which a person called the “potential candidate” chooses whether or not to run in an election in which one other person – called the “current candidate” – is already running. In that model, the potential candidate would win with certainty if she decided to run. That’s unrealistic. Now that you know how to model uncertainty, however, you can fix it! Suppose that if the potential candidate runs, one of two things can happen: She can either win the election or lose the election. Write a probabilistic model depicting the uncertainty the potential candidate faces about whether she will win the election if she runs.

Prompt 2B

The model in Lesson 3 depicted an election in which only one or two persons would be candidates – the “current candidate” and the “potential candidate”. But in many elections, there are more than two persons who consider running, and thus any number of persons who might end up as candidates. Consider a model in which there are three persons – labeled L, M, R who have ideal points located relative to one another like this:

three potential candidates

Suppose that it is uncertain which combination of those three persons will run as candidates in the election. Specifically, any one of the following subsets of these persons might enter the race as candidates: L alone, M alone, R alone, L and M, L and R, M and R, or L and M and R. (Notice that this assume that there is no possibility in which none of the three persons run.)

Write a probabilistic model depicting uncertainty about which of those subsets of the three person will run as candidates in the election. In specifying the probability distribution, assign probabilities so that:

  1. The event in which all three run is less likely than any event in which two of the three persons run;
  2. Any event in which only one person runs is less likely than any event in which two of the three persons run;
  3. Given any pair of distinct events in which two of the three persons run (for instance, the pair consisting of the event in which L and M run and the event in which M and R run), if one contains M and the other doesn’t, the one containing M is more likely than the other.

Rubric

A valid probabilistic model of uncertainty consists of two elements:

  1. An exhaustive list of mutually exclusive events.
  2. A valid probability distribution over those events.

Further, given an exhaustive list of mutually exclusive events, a valid probability distribution assigns a number between 0 and 1 to each of those events, with the numbers assigned to the full list of mutually exclusive events summing to 1.

A valid probabilistic model of uncertainty represents a given uncertain situation if its list of events correspond exactly to the possible events that can occur in that situation, and if the probability distribution in that model assigns probabilities that are ordered and scaled relative to one another in a way that corresponds to the ordering and relative scale of the likelihoods of the different events in the uncertain situation.

Thus your response to each prompt will earn up to 2 points as follows:

  • 2 points if you respond with a valid probabilistic model that represents the uncertain situations described in the prompt.
  • 1 points if you respond with a valid probabilistic model that represents the uncertain situation described in the prompt in all respects except the required ordering or scaling of the probabilities.
  • 0 points if what you write (a) is not a valid probabilistic model OR (b) is a valid probabilistic model but mis-represents the situation described in the prompt in any respect other than the required ordering or scaling of the probabilities (whether or not it correctly orders and scales the probabilities).