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?