3.3 Probability, Conditional Expectations, and Bayes

Probability, Conditional Expectations, and Bayes

Probability assigns numerical weights to uncertain outcomes. In Level I Quant, the point is practical: expected return, scenario analysis, risk forecasting, credit events, and manager selection all rely on probability rules. Keep events, outcomes, and conditions separate on your scratch paper.

The probability of an event ranges from 0 to 1. The complement rule is P(A') = 1 - P(A). The addition rule for either event is P(A or B) = P(A) + P(B) - P(A and B). If events are mutually exclusive, P(A and B) = 0, so the subtraction term disappears.

Joint probability is the probability that two events both occur. Conditional probability is the probability of one event given another: P(A|B) = P(A and B) / P(B). The multiplication rule follows: P(A and B) = P(A|B)P(B). If A and B are independent, P(A|B) = P(A).

Mutually exclusive and independent are different ideas. Mutually exclusive events cannot occur together. Independent events can occur together, but one event does not change the probability of the other. Two events with positive probabilities cannot be both mutually exclusive and independent.

Expected value is a probability-weighted average: E(X) = sum p_i x_i. For returns, it is expected return. For a random variable, variance is sum p_i [x_i - E(X)]^2. The expected value of a decision should be paired with risk, because two choices can have the same expected value and very different dispersion.

Conditional expected value weights outcomes after a condition is known. For example, expected equity return during an expansion can differ from expected equity return during a recession. Analysts often build probability trees with economic states in the first branch and asset returns or default outcomes in later branches.

The total probability rule combines conditional branches: P(A) = P(A|B)P(B) + P(A|B')P(B'). With multiple states, sum across all states. This rule is a common way to move from state-specific default probabilities to an overall default probability.

Bayes' formula reverses the conditioning after new evidence arrives. In two-state form, P(B|A) = [P(A|B)P(B)] / P(A), where P(A) often comes from total probability. The prior is the original probability of B, and the posterior is the updated probability after observing A.

A clean tree prevents most errors. Put the first event on the left, conditional second events on the right, and multiply along branches. Add branch products to get unconditional probabilities. When asked for a conditional probability, divide the relevant joint probability by the probability of the condition.

Bayes questions reward labeling. If a signal is accurate 80% of the time and a condition is rare, the posterior probability may still be modest. The base rate matters. In investment work, ignoring base rates can lead analysts to overreact to a model signal, credit warning, or screening result.