Charta is on Google Play: practise on the go. Get the app
Charta

Use the probability rules for addition and multiplication

CFA® Level I · Quantitative Methods · Statistical Distributions for Financial Asset Prices and Returns · QM.4.a

The idea

The joint probability of two events is P(AB) = P(A|B) x P(B): the chance B happens, then the chance A happens given it did. Independence is the special case where P(A|B) = P(A), so the joint probability collapses to the product of the two unconditional probabilities. For either event occurring, P(A or B) = P(A) + P(B) - P(AB), and the subtraction is the part people drop — without it, outcomes belonging to both events are counted twice. Most errors in this material come from assuming an independence the question never stated.

Formulas

Addition rule of probability

P(A or B) = P(A) + P(B) − P(AB)

A, B
the two events
P(AB)
the probability that both happen

Always. For mutually exclusive events P(AB) is zero and the subtraction drops out — which is the special case, not the rule.

Multiplication rule of probability

P(AB) = P(A | B) × P(B)

A, B
the two events
P(A | B)
the probability of A given that B has happened

Always. For independent events P(A | B) = P(A), so it reduces to P(A) × P(B).

A practice question

From the question bank, with the reason each wrong option is wrong.

Two events are independent when the occurrence of one:

  1. makes the other impossibleWhy not: That is mutual exclusivity, which is the opposite of independence: learning that one happened tells you the other did not.
  2. guarantees the otherWhy not: That is the strongest possible dependence — P(A given B) = 1, when independence requires P(A given B) = P(A).
  3. does not change the probability of the other — correct

Independence is a statement about conditional probability: P(A given B) is simply P(A).

A harder one

A fund beats its benchmark in 60% of years. Assuming independence, the probability it beats the benchmark in two consecutive years is:

  1. 0.60Why not: The probability for a single year. Requiring it twice can only lower the chance.
  2. 0.36 — correct
  3. 0.30Why not: 0.60 halved. Two independent events are combined by multiplying their probabilities, not by dividing one of them.

Independence lets the joint probability collapse to the product: 0.60 x 0.60 = 0.36.

The hardest tier

Two bonds each default with probability 0.10. If the defaults are positively correlated rather than independent, the probability that both default is:

  1. exactly 0.01Why not: That is the independent case, 0.10 x 0.10. Positive correlation is precisely the departure from independence.
  2. above 0.01 — correct
  3. below 0.01Why not: The right departure, the wrong direction. Positive correlation means one default makes the other more likely, which lifts the joint probability.

Independence gives 0.01; positive correlation means one default raises the chance of the other, lifting the joint probability.

In the app: 18 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.

Take the free diagnostic →

More in Statistical Distributions for Financial Asset Prices and Returns