11.1 A Philosophical Approach to Probability » Test Your Knowledge: Explanations
Q1. Who developed the mathematical formulation of probability theory that we still use today?
A. Kolmogorov
B. Pascal
C. Probabilicus
D. Leplace
Correct answer: (A) The Soviet mathematician Andrey Kolmogorov's rigorous formalization of probability theory, completed in 1933, still underlies all of modern probability theory. Blaise Pascal and Pierre-Simon Leplace were French mathematicians who predated Kolmogorov and did important early work on probability, but their formalization are no longer taken to define probability itself. "Probabilicus" is a made-up name.
Q2. In probability theory, what is the "power set" of the sample space?
A. It is the set containing no elements.
B. It is the set containing all subsets of the sample space.
C. It is the set containing all elements not in the sample space.
D. All of the above.
Correct answer: (B). The power set of any set, including the sample space used in probability theory, is the set of all subsets of that set. Option 1 describes the empty set, which is an element of the power set, but is not the entire power set. Option 3 is the complement of the sample space, which is distinct from the power set. Recall that in probability theory, the sample space is the space of all possible outcomes of a process. Thus, the power set of the sample space is the set of all sets of possible outcomes.
Q3. Which of the following accurately describes the frequency interpretation of probability?
A. The probability of a set of possible outcomes is a measurement of the propensity of any element in that set of outcomes to occur.
B. The probability of a set of possible outcomes is a measurement of some agent's subjective degree of belief that an outcome in that set will occur.
C. The probability of a set of possible outcomes is the frequency with which any element of that set occurs in the limit of an infinite long run of trails.
D. None of the above.
Correct answer (C). The third option correctly describes the frequentist interpretation of probability, according to which the probability of a set of outcomes is just the frequency with which elements of that set occurs in a long-run sequence of trials. The first option, by contrast, described the propensity interpretation of probability, while the second describes a subjective, or Bayesian interpretation of probability.