This document summarizes three philosophical interpretations of probability: classical probability, frequency probability, and subjective/inductive probability. Classical probability defines probability as the ratio of favorable to possible outcomes assuming equal possibility. Frequency probability defines probability as the limit of relative frequencies of outcomes from repeated trials. Subjective probability defines probability as a person's degree of belief based on reason and experience, but it is difficult to justify inductive inferences.

1. Philosophical
Interpretations
of Probability
K a y l a L e u n g
A P S t a t i s t i c s F i n a l P r o j e c t

2. Classical Probability
Pierre-Simon Laplace
“The probability of an event is the ratio of the number of cases favorable to
it, to the number of all cases possible when nothing leads us to expect that
any one of these cases should occur more than any other, which renders
them, for us, equally possible.”
You roll a dice. What is the probability that you get an even number? 3/6 where
P(A) = number of favorable cases / total number of equally possible cases.
Principle of Indifference: We should treat outcomes as equally possible if we
have no reason to consider one outcome more probable than the other i.e.
assumed systems of symmetry.
Classical probability is limited to finitely rational possible outcomes. No
possibility for infinite or irrational probabilities.
Criticisms
Classical Probability relies on the assumption that the different outcomes are
equally possible. What does it mean for events to be "equal"? It seems that our
interpretation of probability invokes the notion of probability.

3. Frequency Probability
Consider a coin toss. To define the probability, we toss the coin multiple
times and record relative frequencies. We consider the probability to be
the limit as the number of tosses approaches infinity.
Frequentism is capable of reconciling the finite limitations of classical
probability theory.
How do we necessarily know that the relative frequencies will converge or can
converge upon a finite probability?
Criticisms
Frequency probability struggles to deal with single case probability in that it makes
claims about results viewed in the sequence of thousands or infinite trials, but not
single cases.
Difficulties also arise with forecast predictions such as weather, earthquakes, etc...
in that the do not occur in a pattern that fits in with the frequentist approach.

4. Subjective/Inductive Probability
P(A) = degree of belief that A is true, based on reason, intuition, estimates
Inductive experiential inferences with true premises often have true
conclusions, although not always.
Criticisms
We cannot justify the claim that any inductive inference with true premises
will have a true conclusion regardless of any seemingly probable likelihood
since every inductive inference might have a false conclusion
We are not justified in assuming that because the past has resembled
present that it will continue to do so in the future.
The Sunrise Problem: What is the probability that the sun will rise
tomorrow?
As humans, we constantly make mistakes in terms of our guesses and
estimations, why should we trust our own judgement.