3.3 Probability, Conditional Expectations, and Bayes

Probability, Conditional Expectations, and Bayes

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

Core probability rules

Probabilities lie between 0 and 1. Empirical probability comes from historical frequency, a priori probability from formal reasoning, and subjective probability from informed judgment. The complement rule is P(A') = 1 - P(A). The addition rule is P(A or B) = P(A) + P(B) - P(A and B); for mutually exclusive events the joint term is zero. Odds for an event are P/(1-P), and odds against are the reciprocal.

Joint probability is the chance both events occur. Conditional probability is P(A | B) = P(A and B)/P(B). Rearranging gives the multiplication rule P(A and B) = P(A | B) x P(B). If A and B are independent, P(A | B) = P(A), so the joint probability is simply the product P(A) x P(B). A crucial distinction: mutually exclusive events cannot occur together (joint = 0), whereas independent events can co-occur but neither shifts the other's probability. Two events with positive probabilities cannot be both mutually exclusive and independent.

Expected value, variance, and covariance

Expected value is a probability-weighted average: E(X) = sum p_i x_i. For returns it is expected return. The variance of a random variable is sum p_i [x_i - E(X)]^2. Two choices can share an expected value yet differ sharply in dispersion, so always pair the expectation with a risk measure. Covariance from scenarios multiplies each asset's deviation from its own expected return, weights the products by state probability, and sums: positive covariance means the assets tend to beat their averages together.

Conditional expectations and total probability

Conditional expected value weights outcomes after a condition is known; expected equity return during an expansion can differ from that during a recession. Analysts build probability trees with economic states in the first branch and asset returns or default outcomes in later branches. The total probability rule combines the branches: with two states P(A) = P(A | B)P(B) + P(A | B')P(B'), and with many states you sum P(A | S_i)P(S_i) over all states. This is the standard route from state-specific default probabilities to an overall default probability.

Bayes' formula

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

Counting and exam tactics

Three counting rules appear: the factorial n! orders n items; permutations nPr = n!/(n-r)! choose r ordered from n; and combinations nCr = n!/[(n-r)! r!] choose r when order does not matter. A clean tree prevents most errors: put the first event on the left, conditional events on the right, multiply along branches, and add branch products for unconditional probabilities. For a conditional probability, divide the relevant joint probability by the probability of the condition.

Worked Bayes example

Return to the credit model that flags 70% of future defaulters and 20% of survivors, with a 10% prior default rate. Total probability gives P(flag) = 0.70(0.10) + 0.20(0.90) = 0.07 + 0.18 = 0.25. The posterior probability that a flagged firm actually defaults is P(default | flag) = P(flag | default) x P(default) / P(flag) = (0.70 x 0.10)/0.25 = 0.07/0.25 = 28%. Even though the model catches 70% of defaulters, a flagged firm has only a 28% chance of defaulting, because survivors vastly outnumber defaulters and contribute most of the flags.

This is the base-rate lesson the exam loves to test: high signal accuracy does not translate into a high posterior when the underlying condition is rare.

Counting refresher and exam tactics

Use the labeling formula n!/(n1! n2! ... nk!) when assigning n items into k labeled groups (for example, rating analysts into buy/hold/sell buckets). Bayes questions reward labeling each probability before plugging in. When a stem describes a test, screen, or signal with a stated accuracy and asks for the probability of the underlying condition, recognize it as a Bayes problem and build the denominator with total probability first. When events are described as independent, never subtract a joint term you have already accounted for, and confirm independence by checking whether P(A and B) equals P(A) x P(B).

Ignoring base rates leads analysts to overreact to a model alert, a credit warning, or a screening result.