Probability

Section: Probability Concepts Estimated study time: 45 minutes Content: Probability theory provides the mathematical framework for quantifying uncertainty in investment analysis. The probability of an event is a number between 0 and 1 representing the likelihood that the event occurs. Unconditional probability (marginal probability) is the probability of an event regardless of any other events: P(A). Conditional probability is the probability of an event given that another event has already occurred: P(A|B) = P(A and B) / P(B). This distinction is fundamental — the probability that a stock earns a positive return is different from the probability that it earns a positive return given that the market has declined. For independent events, P(A|B) = P(A): knowing that B occurred provides no information about A. The multiplication and addition rules govern the calculation of joint and combined probabilities. The multiplication rule for joint probability states: P(A and B) = P(A|B) × P(B). For independent events, this simplifies to P(A and B) = P(A) × P(B). The addition rule states: P(A or B) = P(A) + P(B) – P(A and B). For mutually exclusive events (which cannot both occur), P(A and B) = 0, so P(A or B) = P(A) + P(B). Bayes' theorem is a critical extension: it allows updating of prior probabilities based on new information. The formula is: P(A|B) = [P(B|A) × P(A)] / P(B). In investment contexts, Bayes' theorem is used to update the probability of a scenario (e.g., recession) given observed economic data (e.g., yield curve inversion). Expected value is the probability-weighted average of all possible outcomes: E(X) = Σ[P(xi) × xi]. For a portfolio, expected return is the weighted sum of expected returns of individual…