Consider two independent events, A and B. We would say the probability of A and B occurring is:

P(A ∩ B) = P(A) * P(B)

However, what if A is considered a zero probability event? Not an impossibility, but a zero probability event? (Not the empty set)

If this is the case, what can we say about the relationship between P(A) and P(A ∩ B)?

Does P(A) = P(A ∩ B), because P(A) = 0? Or, is P(A ∩ B) < P(A)?

B is not a zero probability event:

P(B) ∈ (0,1)

Since $A\cap B \subset A$, we have $P(A\cap B) \leq P(A) = 0$. The relationship $P(A\cap B) = P(A)P(B)$, but it doesn't give anything useful about $B$. Independence with almost-never-occurring events isn't a very useful concept, since the first equation in your post holds trivially. (If you state independence terms of conditional probabilities, note that $P(B\vert A)$ is't defined for $P(A) = 0$).

  • $\begingroup$ So, P(A∩B)≤P(A)? But, not P(A∩B)=P(A)? $\endgroup$ – Nate Jan 17 '16 at 6:08
  • $\begingroup$ Well, $P(A) = 0$. $\endgroup$ – anomaly Jan 17 '16 at 6:15

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