# Does independence and mutual exclusivity induce impossibility?

Given that we know A and B are independent and they never occur at the same time, one of them must be impossible, no? $$P(A\mid B)=\frac{P(A \cap B)}{P(B)}\\ \text{if A and B independent, B gives no additional information on the probability of A}\\ P(A\mid B) = P(A) = \frac{P(A \cap B)}{P(B)} \Rightarrow P(A) * P(B) = P(A \cap B)$$

If they then never occur at the same time, meaning $$P(A \cap B) = 0$$, either $$A$$ or $$B$$ must be impossible or not?

Edit: I know of the basics of conditional probability as used here. This is not part of any homework, just engagement with this on my own, to refresh my stats knowledge in combination with a learning about data analytics (Bayes' classifier).

• I might be wrong but, isn't being independent and "not occurring at the same time" a contradictory statement? May 8, 2019 at 12:54
• Please add the [self-study] tag & read its wiki. Then tell us what you understand thus far, what you've tried & where you're stuck. We'll provide hints to help you get unstuck. May 8, 2019 at 13:40
• @gung Added the self-study tag. I'm not really doing this for any class. And simply wanted to ask this question, as I saw it as odd. I'm not really doing this for class, but rather as part of a private endeavour on DataCamp. I have tried what I wrote and understand what I wrote, the basics of conditional probability. Would you mind removing the hold tag, so I can get some answers. May 8, 2019 at 14:21
• @ffppnnr You're almost, but not quite, correct. Consider the infinite sequences of iid Bernoulli$(p)$ variables. The set of sequences in which exactly one result is $1$ and the set of sequences in which exactly one result is $0$ obviously are disjoint ("not occurring at the same time") yet, by definition, are independent, because each event has probability zero and so does their intersection. I chose this example to emphasize that probability-zero events are not necessarily "impossible."
– whuber
May 8, 2019 at 14:24

Correct. There is a fundamental contradiction here, and that lies in the fact that mutually exclusive (or incompatible) events cannot be independent, unless, as you suggest, we are dealing with events of measure $$0$$.
• Could you explain what this 'measure of $0$' means ? May 8, 2019 at 14:32
• It means that the probability measure of an event $A$ is $0$ i.e. $P(A)=0$. In probability jargon, you will also hear the terms almost impossible or an event happening almost never. For further detail, I defer you to the excellent Wikipedia article. May 8, 2019 at 19:25