Probability as Long-Term Frequency
According to this definition, the probability of an event is equal to the long-term frequency of the event’s occurrence when the same process is repeated many times.
Under the Long-Term Frequency definition, the probability of rolling an odd number is 0.5 because rolling the die many times leads to roughly half of the rolls being odd.
This definition is different from the classical in two major ways:
- It doesn’t refer to the principle of indifference. More generally, probabilities of single events aren’t determined in advance by conceptually analyzing the sample space
- Probabilities are not defined for single trials. This is a very important distinction that separates this definition from the others. There is no probability of rolling an odd number in a SINGLE roll. The true probability is equal to the percentage of “odd number” outcomes from a hypothetical series of an infinite number of rolls.
A consequence of the second point is that probabilities only make sense when assigned to repeatable events:
- rolling a die
- flipping a coin
- drawing a card from a deck
- measuring the position of an electron in an atom
- etc
Probability as Long-Term Frequency - Limitations
With this definition, you don’t calculate probabilities of particular hypotheses. Each hypothesis is either true or false and probabilities don’t quantify any uncertainty.
This is a counter-intuitive aspect of this definition that many people struggle with because we’re all used to saying things like “it’s unlikely that X is true”, implying that the probability for X to be true is low. But these kinds of statements are meaningless if you use the long-term frequency definition.