Three Views of Probability

"What is the probability of $X$?" hides a question about interpretation. There are three standard answers, and they are not rivals so much as tools for different situations.

Classical

Count the outcomes, assume they are equally likely, divide:

This is the view behind dice, coins and cards, and behind nearly every combinatorial interview question. Its strength is that it needs no data at all. Its weakness is the equal-likelihood assumption, which is doing all the work and is usually false outside a casino.

Getting the counting right is the entire game here, which is why counting methods get their own section.

Frequentist

Probability is the long-run relative frequency of an event over repeated trials. The probability that a market-making strategy wins on a given trade is the fraction of trades it wins, measured over many trades.

This is the view behind backtesting, hit rates and most of classical statistics. Its strength is that it is grounded in observation. Its weakness is that it needs repetition, and it is silent on one-off events. "What is the probability this firm hires me?" has no frequentist answer, because the trial does not repeat.

Bayesian

Probability is a degree of belief, which starts somewhere and updates as evidence arrives:

You hold a prior belief $P(H)$, observe evidence $E$, and end with a posterior $P(H \mid E)$. This is the view behind pricing under incomplete information, and it is how a market maker actually thinks: you have a fair value, order flow arrives, you update.

Its strength is that it handles one-off events and lets you use prior knowledge. Its weakness is that the prior is a judgement call, and a bad prior takes a lot of evidence to overcome. The full treatment is in Bayesian inference.

Which one is a question asking for?

The three views usually agree on the numbers. Where they differ is what question they are equipped to answer.

• Structure is known and outcomes are symmetric, use classical.
• The situation repeats and you have data, use frequentist.
• The event is one-off, or you have prior information and evidence arriving, use Bayesian.

Trading uses all three within a single decision. You might price an option with a classical model of the underlying's outcomes, validate it against a frequentist study of historical fills, and then update your fair value Bayesian-style as order flow arrives during the session.