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Characterise mutually exclusive events and independent events.

FRM® Part I · Quantitative Analysis · Fundamentals of Probability · FRM1.QA.1.b

The idea

Two events are mutually exclusive when they cannot occur together, so P(A and B) is zero and the union has probability: Union = P(A) + P(B). Two events are independent when learning that one occurred does not change the probability of the other. For independent events the joint probability is the product of the two probabilities: Joint = P(A) × P(B). Equivalently, P(A | B) is just P(A). For independent events the union is the sum minus the overlap: Union = P(A) + P(B) − P(A) × P(B). The two ideas are close to opposites. If both events have positive probability and are mutually exclusive, they are dependent, because knowing A happened forces the probability of B to zero. Likewise, independent events with positive probabilities can always occur together. The usual mistake is to treat mutually exclusive as meaning unrelated, or to add probabilities for events that overlap. On the exam, check which condition the question actually states before choosing the formula.

Formulas

Probability of either of two independent events

P(A or B) = P(A) + P(B) − P(A) × P(B)

P(A)
the probability of event A
P(B)
the probability of event B

Only for independent events. If the events are mutually exclusive the overlap is zero instead, and the union is just P(A) + P(B).

A practice question

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

Events A and B are mutually exclusive. Which statement must be true?

  1. P(A and B) = P(A) x P(B), because the events do not influence each otherWhy not: Confuses mutual exclusivity with independence; the product rule defines independence.
  2. P(A) + P(B) = 1, because one of the two events must occurWhy not: Confuses mutually exclusive with exhaustive; neither event need occur, so the probabilities need not sum to one.
  3. P(A or B) = P(A) + P(B) - P(A) x P(B)Why not: Applies the union formula for independent events; for mutually exclusive events the overlap is zero.
  4. A and B cannot both occur, so P(A and B) = 0 - correct

Mutually exclusive means the events have no outcome in common: the joint probability is zero and P(A or B) = P(A) + P(B). Independence and exhaustiveness are different properties.

A harder one

Events A and B each have positive probability and are mutually exclusive. Which conclusion is correct?

  1. They are independent: since they cannot occur together, neither can influence the otherWhy not: Treats 'cannot occur together' as 'no influence'; but the occurrence of A rules B out, which is the strongest form of influence.
  2. They are independent only if P(A) = P(B)Why not: Equal probabilities do not create independence; P(A and B) = 0 never equals a positive product P(A) x P(B).
  3. They are dependent: once A has occurred, the probability of B falls to zero - correct
  4. They may be independent or dependent; mutual exclusivity places no restriction on dependenceWhy not: With positive probabilities, P(B | A) = 0 differs from P(B) > 0, so dependence is certain.

Independence needs P(A and B) = P(A) x P(B) > 0, but mutual exclusivity gives P(A and B) = 0. Hence two mutually exclusive events with positive probabilities are always dependent.

The hardest tier

Two unrelated issuers default over the year independently, with probabilities of 4% and 6%. What is the probability that exactly one of them defaults?

  1. 9.52% - correct
  2. 10.00%Why not: Adds the two probabilities, which counts the case where both default twice.
  3. 9.76%Why not: Computes the probability that at least one defaults, which includes the both-default case.
  4. 0.24%Why not: Computes the probability that both default.

Exactly one default: 0.04 x 0.94 + 0.96 x 0.06 = 0.0376 + 0.0576 = 0.0952, or 9.52%. The sum 10% overstates this by twice the joint probability 0.24%.

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