Short Answer Questions (submit online)

1. What is an interpretation of probability? Why does probability need an interpretation?
2. Explain the difference between a classical interpretation and a frequency interpretation of probability.
3. In what ways does the subjective interpretation of probability differ from the others?

For Further Discussion

• Interpreting probabilistic facts. You have now seen several interesting probabilistic facts, such as 1) The law of large numbers; 2) Conditional probability; and 3) Probabilistic independence. You have also seen several interpretations of probability: 1) the classical interpretation; 2) the subjective interpretation; 3) the frequency interpretation; and 4) the propensity interpretation.
  • Did you find any of these interpretations more plausible than the others? If so, why?
  • These properties can be used to describe many different kinds of systems, from dice to cards to the human genome. How is it possible that such different systems share such specific mathematical properties?
  • As a specific example, suppose someone asks why the average sum of a thrown pair of regular dice is 7. You might answer that it's because of the law of large numbers, and therefore because of probability theory. But suppose they asked further why these mathematical tools apply to things like dice. What could you say to such a person?
  • Relatedly: if someone asks why the laws of gravity apply to both apples falling from trees and skydivers falling from airplanes, the answer might be, "Because both have mass, and the laws of gravity apply equally to all things with mass." But suppose someone were to ask why the laws of probability apply to throwing dice, dealing cards, and interpreting the human genome. How would you answer?
• The classical interpretation. The classical interpretation says that probabilities apply to sets of events of the same kind that can be reduced to cases that are equally possible, like each side of a coin in acoin toss.
  • How might we flesh out what it means to be "of the same kind"?
  • How about "equally possible"?
  • Is possibility the kind of thing that can be treated as a matter of degree, or is everything strictly possible or impossible?
  • What are your thoughts about this interpretation?
• The subjective interpretation. This is an epistemic interpretation of probability, which says that probabilities match the degree of belief of an ideal rational agent.
  • Such an ideal agent may or may not exist. Is this relevant for the interpretation?
  • Following the laws of probability are by far the best way to avoid losing money when gambling. Does this imply that rational agents have beliefs that are probabilistic?
  • On this interpretation, probability is the assignment of a degree of belief between 0 (belief that an event definitely will not happen) and 1 (belief that an event definitely will happen). What number should we apply when we simply have no opinion on the matter? Or when we are completely ignornat? Does this present any particular challenge to the interpretation?
• The frequency interpretation. This interpretation says that the probability of X with respect to a reference class Y is the relative frequency of actual occurrences of X in Y.
  • What should count as a "reference class" on this scheme?
  • Relative frequencies only approximate probabilities in most cases. Is this a problem for the view? Why or why not?
  • Relative frequencies seem to give non-intuitive results for events that only occur once, such as a coin that flips one and is then destroyed. What does the relative frequency theory say about that? Is this a problem? Why or why not?
• The propensity interpretation. Here we say that probabilities are a habit or tendency of things. What does it mean to be a habit or tendency? Is it possible to say this without begging the question, or else reducing this view to one of the other interpretations? It has been suggested that propensities are time asymmetric. How exactly does that suggestion work, and is it a problem?