Probability

Probability has very few rules. Nearly every mistake made with it is the same mistake: confusing $P(A \mid B)$ with $P(B \mid A)$.

The rules

A probability is a number in $[0, 1]$, and the probabilities of all possible outcomes sum to 1.

Addition. For either of two events:

$P(A \cup B) = P(A) + P(B) - P(A \cap B)$

The subtraction is there because adding both counts the overlap twice. When the events cannot both happen, the overlap is zero and the rule simplifies — but assuming that without checking is a common error.

Multiplication. For both:

$P(A \cap B) = P(A) \times P(B \mid A)$

Conditional. Rearranged, that is the definition:

$P(A \mid B) = \frac{P(A \cap B)}{P(B)}$

Read it as: restrict attention to the worlds where $B$ happened, and ask what fraction of those also had $A$.

Independence, and why it is rarer than assumed

$A$ and $B$ are independent when $P(A \mid B) = P(A)$ — knowing one tells you nothing about the other. Then and only then does $P(A \cap B) = P(A)P(B)$.

Independence is an assumption, not a default. It is also the assumption that fails exactly when it matters:

• Twenty stocks in a portfolio look independent in calm markets and move together in a crash. Chapter 6 of the Risk subject is entirely about this.
• Daily returns are approximately independent, which is what licenses the $\sqrt{t}$ scaling in chapter 3 — and chapter 13 shows where that approximation breaks.
• Loan defaults in a portfolio are independent until the thing causing one default causes the others.

A model that multiplies probabilities has assumed independence whether or not anybody said so.

Bayes' theorem

From the multiplication rule written both ways:

$P(A \mid B) = \frac{P(B \mid A) \, P(A)}{P(B)}$

This is not a deep result — it is two lines of algebra. It is important because it is the correct way to update a belief when evidence arrives, and because human intuition about it is reliably wrong.

The screening problem, counted rather than computed

Take 100,000 transactions, with 1 in 1,000 fraudulent.

	Fraud (100)	Honest (99,900)	Total flagged
Flagged	99	999	1,098
Not flagged	1	98,901

The test is 99% accurate in both directions. Yet of the 1,098 flagged transactions, only 99 are fraud:

$P(\text{fraud} \mid \text{flagged}) = \frac{99}{1098} = 9.0\%$

A flagged transaction is more than 90% likely to be honest. Nothing is wrong with the test. The honest population is so much larger that 1% of it still swamps 99% of the small one.

This is the base rate fallacy, and the lesson generalises well beyond fraud screening:

• A "sell signal" that is right 70% of the time, applied to a market that rises 70% of the time, is not informative.
• A screen that finds companies with a characteristic shared by most companies has found nothing.
• Chapter 9 of the Technical analysis subject is this chapter applied to chart patterns.

Probability and frequency are not the same thing

A one-in-a-hundred daily event is not an event that happens once every hundred days on a schedule. It is an event with a 1% chance each day, which means a 63% chance of occurring at least once in a hundred days — $1 - 0.99^{100}$ — and a real chance of occurring twice in a week.

The arithmetic that matters:

$P(\text{at least once in } n) = 1 - (1-p)^{n}$

Applied to investing: SEBI's measurement of individual traders in the equity derivatives segment found 91.1% made net losses in FY25. That is not a statement that any one trade is near-certain to lose. It is the compounded result of a small per-trade disadvantage repeated, which is what chapter 10 of the Derivatives subject means when it says costs decide the outcome.

The point

Probability has three rules and one hard idea: conditioning. $P(A \mid B)$ and $P(B \mid A)$ are different numbers, and treating them as the same is the base rate fallacy — which is why a 99% accurate test on a rare event produces mostly false positives. Independence is an assumption that quietly enters any model that multiplies probabilities, and it fails precisely when the correlation would have mattered.