Conditional Probability

Conditioning is a change of denominator. The unconditional probability of A measures its weight against the entire sample space; the conditional probability measures the same joint outcome against the reduced space in which B is known to have occurred. Dividing the joint probability by the probability of the conditioning event rescales the surviving outcomes so they once again sum to one. This is why P(B) must be strictly positive: conditioning on an impossible event leaves nothing to rescale.

Two consequences follow directly. First, a conditional probability can be larger or smaller than the unconditional one, depending on whether learning B concentrates or disperses the probability mass over A. Second, if the two are equal, the events are independent, because knowing B changes nothing about A. Independence is therefore a conclusion drawn from the arithmetic, not an assumption a candidate is free to impose.

Rearranging the definition gives the multiplication rule: the joint probability equals the conditional probability multiplied by the probability of the conditioning event. Chaining this across a tree diagram is how scenario problems are built, with each branch contributing one conditional factor.

The total probability rule runs the logic in reverse. If a set of scenarios is mutually exclusive and exhaustive, the unconditional probability of an event is the weighted average of its conditional probabilities across those scenarios, weighted by scenario probabilities. Expected value works identically: the unconditional expectation is the probability-weighted average of conditional expectations.

Bayes' formula then inverts the conditioning. The updated probability of a scenario given an observation equals the likelihood of that observation under the scenario, multiplied by the prior scenario probability, divided by the total probability of the observation. Prior belief times evidence, normalised.

Credit analysis is conditional throughout. A transition matrix reports the probability of migrating to each rating grade given the current grade, and loss given default is itself a conditional expectation, defined only on the branch where default has occurred. Expected credit loss multiplies a default probability by a loss severity that is conditional on that default.

Risk measurement follows the same structure. Value at risk states a loss threshold at a chosen probability; conditional VaR, or expected shortfall, is the average loss given that the threshold has been breached. In equity work, backtested screens report hit rates conditional on a signal firing, and analysts routinely confuse the reliability of a signal with the frequency of the outcome it points to. Portfolio construction treats correlations as conditional on the volatility regime, which is why diversification benefits estimated in calm markets often fail to survive a crisis.

A candidate should be able to move fluently among four quantities: the joint probability, each unconditional probability, and both conditional probabilities. Given any three, the fourth follows. Building a two-by-two contingency table of counts or probabilities is usually faster and safer than manipulating symbols, and it makes the shrinking denominator visible.

Beyond arithmetic, be ready to test independence by comparing a conditional probability with its unconditional counterpart, to apply the total probability rule when a variable depends on an unobserved state such as the economic cycle, and to update a prior after new evidence arrives. Watch the base rate: when the conditioning event is rare, a highly accurate signal can still produce a modest posterior probability, because the small number of true cases is swamped by false positives drawn from a much larger population.