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Part CCSIR NET June 2023symmetry-of-dependence

Symmetry of dependence

Let A, B be two events in a discrete probability space with P(A) > 0 and P(B) > 0. Which of the following are necessarily true?

  1. A.If P(A | B) = 0 then P(B | A) = 0.✓
  2. B.If P(A | B) = 1 then P(B | A) = 1.
  3. C.If P(A | B) > P(A) then P(B | A) > P(B).✓
  4. D.If P(A | B) > P(B) then P(B | A) > P(A).

Solution

(1) P(A∩B) = 0 is symmetric. (3) P(A|B) > P(A) ⇔ P(A∩B) > P(A)P(B), symmetric in A, B. (2) A ⊇ B (a.s.) does not give B ⊇ A. (4) Take B ⊂ A with P(B) small.

The trap it tests

Dependence misread

Independence, pairwise vs mutual, or correlation vs dependence.

Drill statements like this

Related counterexample: Pairwise independent events are mutually independent

From Probability › Axioms, conditional probability, independence, Bayes

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