[Math] Conditional probability in independence and mutually exclusive events.

Question

This thread shows that if two events are to be mutually exclusive and independent, one of them should have zero probability. I worked the following example that seems to contradict conditional probability.

We pick a real number in range [0,1]

A = event that the number is rational

B = event that the number is irrational

P(A and B) = $0$ (because they are disjoint by definition)

Also, P(A) = $0$ (from measure theory/discrete maths)

Thus P(A and B) = P(A) * P(B)

This means A and B are independent.

Note that the probability that number is irrational given that it is rational is zero. This does not satisfy conditional probability.

P(B | A) = 0 $\ne$ 1 = P(B)

P(B | A) $\ne$ P(B)

This just means that occurrence of one event (A) is affecting the probability of other (B), and not independent. Where did I go wrong?

Answer 1

It's fine.   Contradictions are allowed to happen when the precondition is not satisfied.

If $\mathsf P(A)>0$, then $\mathsf P(B\mid A)\cdot\mathsf P(A) = \mathsf P(B)$ .

Even if the measure $\mathsf P(B\mid A)$ can be evaluated from the model, the consequent doesn't have to hold if the antecedent doesn't; ie: $\mathsf P(A=0)$.