Probability

DATE LINE 11/30 OR 12/5

A Dialogue Concerning the Meaning of Probability

For this project you should imagine yourself in conversation with a student who has come to you to learn about the meaning of probability. The student’s previous understanding is that the probability of an event is simply the frequency of occurrence over repeated trials, and your task is to convince them otherwise. Below you will find the student’s half of the dialogue, asking you questions along the way. Fill in the sections marked “[Your response]” with answers to these questions that you think would be convincing. The length of these answers is up to you, but try to keep your arguments concise while completely answering each question. Be sure to provide examples where appropriate.

Student: “I heard there’s a lot of controversy about the meaning of probability, but I’m not sure what all the fuss is about. Isn’t the probability of something just how often it happens over many trials? What’s an example of an application of probability where the frequency interpretation wouldn’t make sense?”

[Your response]

Student: “OK, so if probability isn’t simply just the frequency of occurrence, what is it?”

[Your response]

Student: “Wait a minute. In previous classes I learned that probability has certain rules like P(A or B) = P(A) + P(B) - P(A and B). Those make sense to me if we’re talking about relative proportions. But if probability is just a number that expresses how certain we are about a proposition, why do these rules of probability hold true?

[Your response]

Student: “So what are the rules you’re assuming probability has to satisfy? What’s the justification for those rules?”

[Your response]

Student: “OK, so I’m willing to believe that if we assign probabilities to propositions in a way that satisfies your rules, the ordinary rules of probability can be derived as a consequence. But wouldn’t this mean that probability is always subjective? Can you give an example of a proposition that any two people would have to assign the same probability to?”

[Your response]

Student: “I’ve heard something about this kind of probability being called ‘Bayesian’. Doesn’t Bayes’ Theorem just have something to do with conditional probability? Why is that relevant here?”

[Your response]

Student: “So you’re saying we assign probabilities to propositions based on information. What if those events already happened in the past? Can we still say they have a probability, or is the probability always zero or one depending on whether the event actually happened?”

[Your response]

Student: “OK, let’s say I’m trying to test a hypothesis about something, and I observe some data that would be very unlikely if the hypothesis is true. Can’t I just say the hypothesis is probably false as a result?”

[Your response]

Student: “So far it sounds like probability is just common sense put into numbers. Can you give me an example of a problem I can solve with the rules of probability that I can’t just solve with intuition?”

[Your response]

Student: “Wait a minute. In my Statistics class we learned a lot of tests for different kinds of hypotheses, and I don’t remember hearing anything about Bayes’ Theorem or any ‘prior probabilities.’ Are the tests I learned in my class based on this interpretation of probability or something else?”

[Your response]

Student: “What’s wrong with that way of testing hypotheses? It seems more objective. If it’s so illogical, why have people been doing Statistics that way for so long?”

[Your response]

Student: “OK, but what would be the Bayesian way of doing something like estimating an unknown parameter?”

[Your response]

Student: “Alright, I think I’m starting to get it, but I’m still confused about something. According to your interpretation, probability is a number we assign to propositions that expresses how confident we are in them being true based on background assumptions. But we all know that a coin flip has a 50% probability of coming up heads. How can that be the right answer if probability is so subjective?”

[Your response]

Student: “So does the fact that the frequency of it coming up heads is close to 50% actually matter for the probability?”

[Your response]

Student: “OK, I’m convinced!”