Symmetry of dependence

• A.Correct

  True

  because the hypothesis is secretly symmetric.

  P(A | B) = 0 means P(A ∩ B)/P(B) = 0

  hence P(A ∩ B) = 0.

  That statement makes no distinction between A and B

  so P(B | A) = P(A ∩ B)/P(A) = 0 as well.

  The trigger: rewrite every conditional statement in terms of P(A ∩ B) and see whether what remains is symmetric.

• B.WrongConverse assumed

  False, and here the symmetry genuinely breaks.

  P(A | B) = 1 means P(A ∩ B) = P(B), that is B is contained in A up to null sets.

  Then P(B | A) = P(B)/P(A)

  which is less than 1 whenever P(B) < P(A).

  Containment runs one way only.

• C.Correct

  True, and it is the symmetric case again.

  P(A | B) > P(A) is equivalent to P(A ∩ B) > P(A)P(B)

  which is symmetric in A and B, and that in turn is equivalent to P(B | A) > P(B).

  Positive association is a mutual relation — if B raises the chance of A, then A raises the chance of B by exactly the same criterion.

• D.WrongConverse assumed

  False.

  Written out, the hypothesis is $P(A \cap B) > P(B)^{2}$ and the conclusion $P(A \cap B) > P(A)^{2}$

  which are different statements.

  Take B ⊂ A with P(A) = 9/10 and P(B) = 1/10: then P(A | B) = 1 > 1/10 = P(B)

  so the hypothesis holds

  but P(B | A) = 1/9 (verified) and SymPy confirms 1/9 − 9/10 < 0

  so the conclusion fails.

  Compare with (C): the same shape of statement

  but comparing against the other event's probability destroys the symmetry.

2 of these 2 wrong options correspond to a named reasoning error — the same errors recur across subjects.