What is Probability?

Probability is the mathematical framework for quantifying uncertainty. That definition is easy to nod along to and easy to underrate, so it is worth being concrete about why a trading firm cares.

A market maker quoting a two-sided price is not predicting the future. They are stating a belief about a distribution of outcomes, then charging a spread for the risk of being wrong. Every quote, every hedge and every position size is downstream of a probability estimate. This is why prop firm interviews open with probability rather than with finance: the firm is testing whether you can reason numerically about uncertainty, at speed.

The scale, and what its ends mean

A probability is a number between 0 and 1 assigned to an event:

$P(A) = 0$ means $A$ is impossible, $P(A) = 1$ means it is certain, and everything interesting lives strictly between the two. Both endpoints deserve suspicion. Assigning probability 0 or 1 to anything that is not a logical impossibility is usually a modelling error, and it is the kind of error that produces catastrophic position sizing.

The complement of an event carries the remaining mass:

That identity looks trivial and does an enormous amount of work. Many interview questions are solved fastest by computing the probability of the thing you do not want and subtracting. When a question asks for the chance of "at least one" of something, reach for the complement before you reach for a sum.

From probability to a decision

A probability on its own does not tell you what to do. Pairing it with a payoff does.

Suppose you are offered a contract paying $100 if an event occurs and nothing otherwise, and you believe the event has probability $0.6$. Its expected value is

so $60 is your break-even price. Below it you want to buy, above it you want to sell, and the gap between your estimate and the market's is the entire source of edge. A market maker would quote around that number, perhaps 58 at 62, earning the spread while carrying the risk that the $0.6$ is wrong.

Notice what this framing demands: an estimate, a payoff, and a sense of how confident you are. Those three ingredients are the spine of this course. Probability gives you the estimate, expectation turns it into a price, and variance tells you how wrong it can get.

Why the coins and dice matter

Interview probability is dominated by coins, dice and cards, which can feel disconnected from markets. They are not. They are the cleanest available setting for the reasoning markets demand: enumerate the outcomes, assign mass, compute an expectation, and do it without confusing yourself. A candidate who fumbles conditional probability on a card problem will fumble it on an order-flow problem too, where the numbers are messier and the clock is louder.