1. What a probability is: no certainties

Level 1: Probability: the language of uncertainty

A company is in the middle of a merger. The market puts a 70% probability on the deal closing at the agreed price, and a 30% probability on it breaking and the stock returning to its previous level.

Which of the two prices do you see on the screen? Neither of them.

What you see is a blend of both, weighted by those probabilities. Understanding where that number comes from is the starting point for everything that follows.

A number between 0 and 1

A probability is a number between 0 and 1 assigned to an event to represent how likely it is.

• 0 means impossible.
• 1 means absolute certainty.
• Everything in between expresses different degrees of likelihood.

There is one added rule. When you list all the possible outcomes of a situation that are mutually exclusive — only one of them can happen — their individual probabilities must add up to exactly 1.

This is the formal basis of something the market theory course used informally all the time: calling an outcome likely or unlikely without ever putting a number on it.

Two ways of reading that number

The most classical reading is the frequentist one. The probability of an event is the proportion of times it would occur if the same situation were repeated, under identical conditions, a very large number of times.

It works beautifully for dice, coins and lotteries. There, each repetition is genuinely identical to the last.

Markets rarely offer identical repetitions. Every employment report, every interest rate decision and every quarterly earnings release happens in a slightly different context, with different levels of debt, sentiment and positioning.

That is why it pays to complement it with the subjective or Bayesian interpretation. A probability can also be understood as the rational degree of belief you hold about an event, given all the information available at that moment.

And, above all, it can be revised when new information arrives. This second interpretation is the one that dominates in practice, both in discretionary and in quantitative trading. It is also the direct bridge to the next module, which deals with how that belief is updated in a disciplined way.

The price is not a forecast

This has a very concrete consequence for how you read the price of an asset.

The current price is not a forecast of a single future destination. It is something closer to a probability-weighted average across the whole range of outcomes that market participants consider possible at that moment.

Return to the merger from the opening. With a 70% probability of the deal closing at the agreed price and a 30% probability of it breaking, the current price will reflect that weighted blend, not a binary bet on one of the two outcomes.

An edge is not a guarantee

A very widespread mistake among beginner traders is to treat a technical signal, an indicator or a price target as if it delivered certainty. What they deliver, at best, is a probability shifted away from the starting point.

A well-constructed pattern, verified against data, might raise the probability of an up move from 50% to 55%. The 50% is the starting point with no information at all.

That shift is a real and valuable statistical edge. And it is nowhere near a guarantee.

Confusing a shift in probability with a certainty carries a very concrete price: it leads to completely wrong position sizing and risk management.

Why ten trades prove nothing

This is where the difference between the law of large numbers and what is informally called the law of small numbers comes in.

A real statistical edge only shows up reliably when the process is repeated many times. In a single trade, or even across a run of ten trades, something else is in charge: the random variation inherent to a small sample, not the underlying probability of the strategy.

An uncomfortable conclusion follows. Judging whether a strategy works from one profitable month is statistically uninformative. So is discarding it after one losing month.

It is a mistake the module on expected value revisits in more detail.

Likely and severe are different things

It is also worth separating two questions that tend to get mixed together:

• The probability that an event occurs.
• The magnitude of its consequences if it does.

The fact that the probability of a stock falling more than 50% in a year is relatively low says nothing, on its own, about how much that scenario should worry you.

An unlikely event with an enormous loss attached can weigh as much or more in a decision than a frequent event with a small impact. This combination of probability and magnitude is exactly what expected value formalizes later in this level.

No estimated probability is the final one

One last caution remains, and it is the one that will reappear most often.

No probability computed from market data is the real, definitive probability of anything. It is an estimate, with three limitations worth keeping in mind at all times:

• It is drawn from a limited historical sample.
• It is subject to error.
• It can become invalid if the conditions that generated that data change.

This idea about the stability of estimated probabilities will come back again and again throughout the course, every time backtesting or model building comes up.

All the rest of this level, and to a large extent the whole quantitative mathematics track that follows, is nothing more than a toolkit for one task: turning the vague 'maybe' of everyday language into a number you can compute, compare and put to the test.