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Tell conditional probabilities apart from unconditional ones.

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

The idea

An unconditional probability, also called a marginal probability, is the chance of an event using no additional information, such as the probability that a loan in the book defaults next year. A conditional probability uses extra information, such as the probability of default given that the economy is in recession. They are linked by the law of total probability, which averages the conditional probabilities over the states: Marginal = Σ P(A | Bi) × P(Bi). The sum runs over the events Bi, which are mutually exclusive and exhaustive. The unconditional probability is therefore a weighted average of the conditional ones, with the probabilities of the states as weights. If A and B are independent, then P(A | B) is just P(A), because the information changes nothing. Otherwise conditioning moves the probability up or down. The usual mistake is to use a conditional figure as if it applied to the whole population. On the exam, ask what has been given.

Formulas

Law of total probability

P(A) = P(A | B₁) × P(B₁) + P(A | B₂) × P(B₂) + ...

P(A | B₁), P(A | B₂)
the probability of A in each state
P(B₁), P(B₂)
the probability of each state

The states must be mutually exclusive and exhaustive, so their probabilities sum to one. The unconditional probability is a weighted average of the conditional ones.

A practice question

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

Which of the following is an unconditional probability?

  1. The probability that a loan defaults next year given that it was downgraded last monthWhy not: Conditions on the downgrade, so it is a conditional probability.
  2. The probability that a randomly chosen loan in the book defaults over the next year, with no other information - correct
  3. The probability that a loan defaults next year given that the economy enters a recessionWhy not: Conditions on the recession scenario, so it is conditional.
  4. The probability that the second loan drawn from the pool is in default given that the first drawn was in defaultWhy not: Conditions on the outcome of the first draw, so it is conditional.

An unconditional (marginal) probability uses no extra information. Each of the others adds a given event after the word 'given'.

A harder one

The probability of a recession next year is 25%. A firm's default probability is 8% if there is a recession and 2% if there is not. What is the firm's unconditional default probability?

  1. 3.50% - correct
  2. 5.00%Why not: Takes a simple average of the conditional probabilities rather than weighting by the state probabilities.
  3. 2.00%Why not: Includes only the recession term and omits the no-recession contribution 0.75 x 2%.
  4. 10.00%Why not: Adds the two conditional probabilities without weighting them by the probability of each state.

Law of total probability: 0.25 x 8% + 0.75 x 2% = 2.00% + 1.50% = 3.50%.

The hardest tier

A firm's unconditional default probability is 4%. The probability of a recession is 20%, and the default probability given a recession is 10%. What is the default probability given no recession?

  1. 2.0%Why not: Subtracts the recession term but forgets to divide by the probability of no recession, 0.80.
  2. 3.2%Why not: Scales the unconditional probability by P(no recession), which is not the law of total probability.
  3. 2.5% - correct
  4. 10.0%Why not: Divides by the recession probability 0.20 instead of the no-recession probability 0.80.

Total probability: 0.04 = 0.20 x 0.10 + 0.80 x x. So 0.80 x = 0.02 and x = 0.025, or 2.5%.

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