2. Conditional probability: why context changes everything

Level 1: Probability: the language of uncertainty

An indicator promises that it fires 80% of the times a rally arrives. That sounds close to a certainty.

Once the arithmetic is done, when that indicator lights up, the probability of a rally is barely 23%.

There is no trick in the brochure. The 80% is correct. What is missing is the figure this module is about to put on the table.

Restricting the playing field

Conditional probability is the probability that an event occurs given that another event is already known to have occurred.

The mechanism is easy to describe. Instead of computing the probability over the full universe of possibilities, you restrict the calculation to the subset of cases in which the known condition holds. Inside that smaller subset, you recompute the proportions.

It is, in essence, updating the playing field before asking again how likely something is.

The base rate and the conditional can look nothing alike

The gap between an unconditional probability — the so-called base rate — and a conditional probability can be enormous.

An illustrative example. If the S&P 500 closes down on 46% of all sessions, that is the base rate.

Now change the question: what is the probability of a negative close given that the previous session also closed down? The number can be quite different, typically somewhat higher.

The reason is that volatility tends to cluster in time. Turbulent days follow other turbulent days more often than pure independent chance would suggest.

Bayes: three ingredients

Bayes' theorem is, at heart, a precise rule for updating a prior belief when new evidence arrives. It combines three ingredients:

• How likely the hypothesis was considered before seeing the evidence: the prior probability, or prior.
• How likely it is to observe that specific evidence if the hypothesis were true.
• How likely it is to observe that same evidence in general, whether or not the hypothesis is true.

The updated probability comes from combining the prior with the reliability of the evidence. The result is then rescaled, so that it remains a valid probability between 0 and 1.

The indicator's arithmetic

A numerical market example makes this far clearer than any abstract definition.

The three starting figures:

• A certain technical pattern precedes a real rally only 10% of the times it appears. That is the prior probability.
• When the rally does materialize, a confirmation indicator fires 80% of the time.
• That same indicator also fires, as a false positive, in 30% of the cases where no rally follows.

Now the arithmetic, out of 1,000 appearances of the pattern:

• 100 correspond to a real rally and 900 to no rally.
• Of those 100, the indicator fires in 80 cases.
• Of the remaining 900, it fires in 270 cases as a false alarm.
• In total, the indicator fires 350 times.

And here is the point. Only 80 of those 350 firings correspond to a real rally. The probability of a rally given that the indicator has fired is barely around 23%, a long way from the 80% that sounds so convincing in the indicator's brochure.

The base rate error

This example illustrates the base rate neglect error: the human tendency to focus only on the apparent reliability of the new evidence, that 80%, and ignore how rare or common the underlying event was to begin with, that 10%.

It is exactly the same mistake made by any trader who blindly trusts an indicator's advertised hit rate. The prior question is missing: how frequent the phenomenon it claims to detect actually is.

Updating is not a one-off act

Bayes' theorem is not applied once and forgotten. It is a sequential updating process.

Every new data point serves to revise the belief again: every economic report, every earnings release. And the starting point for that revision is the belief already updated at the previous step, not the original belief.

Conceptually, that is how a rigorous discretionary analysis process should work. So should a quantitative system that blends several information sources. Each piece of evidence nudges the probability a little, it does not fix it in one move.

The two questions to keep apart

The practical value of all this, when evaluating or building trading signals, is that it forces you to separate two questions that are easily conflated:

• Does this fact make the outcome more likely than the starting point?
• How rare or frequent was that starting point before the fact was observed?

Ignoring the second question and keeping only the first is precisely the mental shortcut that leads people to overrate indicators that sound very precise but detect infrequent events.