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I have studied some basic probability theory, and I have seen the following definition of conditional probability: $$P(B|A) = \frac{P(AB)}{P(A)}$$ However, the definition states that $P(A) \neq 0$.

I was wondering if there is a more general definition to deal with situations like this. For example, when our experiment involves choosing a real number at random from the interval $[0,10]$, and $A$ represents the event that the chosen number is an integer, while $B$ represents the event that the chosen number is 5. Since in this experiment $P(A) = 0$, we cannot use the definition of conditional probability. Nevertheless, it is intuitively very plausible that $P(B|A) = 1/11.$

Aria
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  • Interesting question! It's relevant for situations like yours in which a probability of zero does not correspond to impossibility. – Zubin Mukerjee Oct 06 '23 at 15:35
  • This is relevant: https://math.stackexchange.com/questions/110112/ – Zubin Mukerjee Oct 06 '23 at 15:43
  • The main idea is to approximate $A$ by events which have positive probability and then take a limit. However, when doing this, you do need to be precise with your interpretation of the result. There is no canonical way to approximate null events by positive probability events, and in fact, the resulting answer is not independent of the choice of approximating events – Andrew Oct 06 '23 at 15:45
  • @ZubinMukerjee The answers there are not correct – Andrew Oct 06 '23 at 15:46
  • @ZubinMukerjee, Thank you for the related link. I will try to use the definition by PDFs to answer this situation in my example. Probably I have to consider X to be the continuous uniform random variable in [0,10], but it would be very kind of you if you could use that method to explain this case in my question anyway. – Aria Oct 06 '23 at 16:22
  • @Andrew, Thank you for your idea, but the problem is for example if you consider $A_\delta$ to be the event of choosing a real number in the neighborhood of integers of radii $\delta$, and for $B_\delta$the neighborhood around 0 of radius $\delta$ we get $2\delta/20\delta$ which is inaccurate and really depends on the way we define these approximations and could easily give a wrong answer, right? – Aria Oct 06 '23 at 16:25
  • Yes there will be issues here, because once you approximate with (symmetric) intervals around the integers, the endpoints will be neglected. On the other hand, there is no a priori reason one should approximate with symmetric intervals, as opposed to only half intervals. It's all essentially arbitrary. As I mentioned before, it's not about what answer you get that's important, it's more important that you interpret the answer exactly as it is, and not try to attach any extra meaning to it. – Andrew Oct 06 '23 at 16:51
  • And it's important to understand that any time you "condition" on a null event, you are doing some sort of approximation and (depending on how careful you are with your writing/notation) suppressing the dependency of the answer on the choice of approximation. When you use conditional density $Y\mid X$, the approximation is already chosen for you, namely $\lim_{h\downarrow 0}P(x\leq X \leq x+h)$. On the other hand, the formula relies on the use of the joint density $f_{X,Y}$, which in general does not exist. In particular, it does not exist here. – Andrew Oct 06 '23 at 16:56
  • In any case, I suggest to ignore the answer by kludg in the post linked by Zubin; it is wrong. – Andrew Oct 06 '23 at 16:59
  • @Andrew, Taking closed half intervals around integers would be the correct way to deal with it, thank you. I agree that it's important to interpret the answer exactly as it is, but shouldn't we have a method to get consistent answers? To be specific, isn't the definition kind of ambiguous if we get different answers when we use different approximating events? There are infinitely many ways to define these approximating events, and it could be possible to be able to get any arbitrary answer if we adjust the approximating events. Isn't there a precise definition using this idea? – Aria Oct 06 '23 at 17:02
  • Let us continue this discussion in chat. – Aria Oct 06 '23 at 17:17
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    You can sometimes define conditional probabilities on events of measure $0$ using conditional expectation, but conditional expectation is technically only defined almost everywhere so you can have situations where two "conditional probabilities" are written the same way but have different values. – Kakashi Oct 07 '23 at 01:41

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