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Distinguish plain independence of events from conditional independence.

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

The idea

Ordinary independence says the joint probability is the product of the two marginal probabilities: Joint = P(A) × P(B). Conditional independence given a third event C says that P(A and B | C) equals P(A | C) × P(B | C). Once C is known, learning A tells you nothing more about B. The two properties are separate, and neither implies the other. Two firms may default independently given the state of the economy, yet unconditionally their defaults are positively dependent, because both become more likely in a recession and the state is shared. The reverse also happens: events that are independent on their own can become dependent once you condition on something both influence. To get the unconditional joint probability, compute the joint probability within each state and weight it by the probability of the state. The common mistake is to multiply the unconditional marginal probabilities. On the exam, never assume independence the question has not stated.

Formulas

Unconditional joint probability of conditionally independent events

P(A and B) = Σ P(A | Sᵢ) × P(B | Sᵢ) × P(Sᵢ)

P(A | Sᵢ)
the probability of A in state i
P(B | Sᵢ)
the probability of B in state i
P(Sᵢ)
the probability of state i

A and B are independent within each state, and the states are mutually exclusive and exhaustive. Multiplying the unconditional marginals instead understates the joint probability when both rise in the same state.

A practice question

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

Which statement about independence and conditional independence is correct?

  1. Conditional independence given C implies that the events are unconditionally independentWhy not: Ignores that C can be a common driver, such as the economic state, that links the events unconditionally.
  2. Unconditional independence guarantees independence given every possible event CWhy not: Independent events can become dependent once you condition on an event that both influence.
  3. Conditional independence given C requires A and B to be mutually exclusive within CWhy not: Mutual exclusivity within C would make the joint probability zero, which is a different and much stronger property.
  4. Events can be conditionally independent given C and yet be dependent unconditionally - correct

The two properties are separate. Defaults that are independent given the economic state are typically positively dependent unconditionally, because the state is shared.

A harder one

The economy is in a good state with probability 0.70 and a bad state with probability 0.30. Given the state, two firms default independently: each has a default probability of 2% in the good state and 10% in the bad state. What is the unconditional probability that both firms default?

  1. 0.328% - correct
  2. 0.194%Why not: Squares the unconditional single-firm default probability, wrongly assuming the defaults are independent unconditionally.
  3. 0.520%Why not: Averages the two state-level joint probabilities with equal weights instead of the state probabilities 0.70 and 0.30.
  4. 4.400%Why not: Reports the unconditional default probability of a single firm, not the probability that both default.

Joint default within each state is the product of the two conditional probabilities. Weighting by the state: 0.70 x 0.02² + 0.30 x 0.10² = 0.00028 + 0.00300 = 0.00328, or 0.328%. The product of marginals would give 0.194%.

The hardest tier

Two fair coins are tossed independently. Let A be 'the first toss is heads', B be 'the second toss is heads' and C be 'at least one toss is heads'. Which statement about A and B is correct?

  1. They are conditionally independent given C, because independence holds under any conditioning eventWhy not: Assumes unconditional independence is preserved after conditioning; here conditioning on C introduces dependence.
  2. They are not conditionally independent given C: P(A and B | C) = 1/3, but P(A | C) x P(B | C) = 4/9 - correct
  3. They are conditionally independent given C: P(A and B | C) = 1/4 = 1/2 x 1/2Why not: Uses the unconditional probabilities 1/4 and 1/2; given C, P(A and B | C) = 1/3 and P(A | C) = 2/3.
  4. They are not conditionally independent because A and B are mutually exclusive given CWhy not: Both heads is possible given C, so A and B are not mutually exclusive; that is a different concept from independence.

Given C there are three equally likely outcomes: HH, HT, TH. P(A and B | C) = 1/3, while P(A | C) = P(B | C) = 2/3, so the product is 4/9, not 1/3. Independent events can be dependent once conditioned.

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