Interpretations of Probability

The Problem

• Someone taking an English exam about one of 20 novels chooses which ones to study. If she chooses two, what is the probability she has picked correctly?
  (Here we are assuming each novel is equally likely to be picked by the examiners.)
  Classical.
• Someone says, "Well, I don't need to rush to get my husband: the train is late 80% of the time."
  Frequency.
• A businessperson says "The odds are 2-1 I will make a profit this year." What does that mean?
  Subjective.
• The issue of one-off events.

Classical

The classical interpretation of probability deals with events that are equally likely. We count the number of possible outcomes, which we will call n, and then say that the likelihood of event e occurring is 1/n.

Probability Rules

• If there are n equally likely possibilities, of which one must occur and s are regarded as favorable, then the probability of success is s/n.
  Example: My team plays well in bad weather, and will likely wind if the weather is bad. It plays tomorrow, and tomorrow there are equally likely chances of a rainy day, a snowy day, or a nice, sunny day. Two of those events represent "success," so the probability of my team winning is 2/3.
• If there are n equally likely possibilities, which one must occur, then each possibility has the probability 1/n.

Using these rules

What are the of drawing two red cards in a row from a normal deck of cards?

Let's use our knowledge from the previous material. The total number of possibilities is 52 choose 2, while the number of favorable possiblities is 26 choose 2. So the odds are 25 / 102.

Why is it not just 1/2 * 1/2? Because once we have drawn are red card, there are now fewer red cards in the deck!
Compare to coin flips.

Problem: A company produces cars. For every 12 cars that go out, they inspect three. Only if there is a problem in one of them do they inspect the rest. What is the probability that a lot will go out with a defective car?

Odds and probabilities:

If the probability of rolling two sixes in a row is 1/36, then the odds for succes are the number of outcomes that result in success (1) versus the number that result in failure (35). Typically, here we would say, "The odds are 35 to 1 against him."

More information

SEP on Classical theory

Frequency

The basic idea here is that the probability of some event e is equivalent to the number of times we would expect to see e if we repeatedly ran some process that sometimes produces e.
If we repeatedly flip a fair coin, in the long run, we expect to see a roughly equal number of heads and tails.
This is related to, but not the same as, The Law of Large Numbers. The frequentist interpretation says that the probability of some event simply is its rate of occurence in the long run, while the Law of Large Numbers relies on there being something else that is the true probability, and that as our sample gets larger, we move towards that true probability.

Using "similar" events to determine frequency

"When something happens just once, the frequency interpretation lives us no choice but to refer to a set of similar situations." -- Freund, p. 50

But just what situations are "similar"?

• Is the American Republic "similar" to the Roman Republic?
• Is rabies in a population of racoons "similar" to rabies in a populations of skunks?
• Is the 2016 presidential election "similar" to the 1964 presidential election?
• Is this particular cancer patient "similar" to other cancer patients?
• Is this team's attempt to come back from a 3-1 defecit "similar" to others team's attempts?

More information

Wikipedia on frequentist propability.

Subjective

What odds would you accept to bet on event a occuring as opposed to event b occurring?

"If a person feels that a to b are fair odds for betting on a success of a given kind, he is, in fact, assigning to such a success a subjective... probability:
p = a / (a + b)

Example: If I bet on the Jets winning this weekend at 3 to 2 odds, I am assigning a 3/5 (60%) chance the Jets will win.

If I am willing to give odds of 100-to-1, I think the likelihood of succes is 100 / (100 + 1) (99%). If I give odds of 1000-to-1, I think likelihood of succes is 1000 / (1000 + 1) (99.9%).

Axiomatic

Axiomatic probability is not an interpretation of probability at all: it deliberately ignores the matter of interpretation and simply tries to set probability theory on a sound axiomatic basis.

The Axioms of Probability

1. The probability of an event must be 0 or greater:
   P(A) ≥ 0 for any event A.
2. The probability of the sample space is 1:
   P(S) = 1
3. If two events A and B are mutually exclusive, the probability that one or the other will occur is the sum of their probabilities:
   P(A &union; B) = P(A) + P(B)

One-off events

"many of the probailities which we use to express our faith in predictions, judgments, or decisions, are simply 'success ratios' that apply to the method we have employed. -- Freund, Introduction to Probability, p. 51