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Related to these questions:

Prove/Disprove probability of 0 or 1 (almost surely) will never change and has never been different

Does an unconditional probability of 1 or 0 imply a conditional probability of 1 or 0 if the condition is possible?

For probabilities between 0 and 1, we can safely assume that new information could change the probability. What about 0 and 1? This seems to be what NNT is answering here ('no probability that is 0 or 1 should ever change.'), but the vocabulary is too difficult for me :(

BCLC
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  • If you remind yourself that all of the above equalities hold $a.s.$ only, you'll be free of confusion that could arise from the fact that $P(A|A)=1$ even if $P(A)=0$. – A.S. Jan 24 '16 at 04:52
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    @ A.S., nice user name ... a.s. – Mark L. Stone Jan 24 '16 at 13:28
  • @A.S. What do you mean? This time I am asking for intuition – BCLC Jan 24 '16 at 14:09
  • Wait I think I get it. Do you mean that probabilities of 0 or 1 remain unchanged @A.S. with new information? So they could theoretically change with new information? So then the CEO in NNT's post is saying something similar to $X \sim Unif(0,1)$ could be equal to $0.5$ in the sense that he is theoretically correct but practically incorrect? – BCLC Jan 24 '16 at 14:20
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    This is relevant here: [How does a Bayesian update his belief when something with probability 0 happened?](https://stats.stackexchange.com/questions/333633/how-does-a-bayesian-update-his-belief-when-something-with-probability-0-happened#424845) – kjetil b halvorsen Jan 28 '22 at 17:10

2 Answers2

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I'll take a shot at an argument clarifying what I think Taleb means.

Reflect on what it would take to say, with absolute confidence, that some event $A$ is sure to happen. I posit that you need

  • Perfect knowledge on which measurements, entities, and variables can influence the outcome. That is, you can classify all variables $B$ (lets stay in binary land for simplicity) such that $Pr(A \mid B) \neq Pr(A \mid -B)$.
  • The values of all variables $B$ above.

If you do satisfy the above condition, then there is nothing more to be known. You have complete knowledge about all influencers of $A$. So once you can say for certainty that $Pr(A) = 1$, there is no possible information you can acquire that could update this belief, everything else must be irrelevant.

Matthew Drury
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  • Thanks Matthew Drury. This makes sense, but something AS said made me think of something. What do you think of [this](http://stats.stackexchange.com/questions/192179/why-does-a-probability-of-0-or-1-remain-unchanged-with-new-information-intuitiv#comment365214_192179)? – BCLC Jan 24 '16 at 14:21
  • Wait is B some collection of binary random variables? As in $P(A|B) = P(A|X_1, X_2, ...)$? – BCLC Jan 24 '16 at 14:32
  • But $P(A)=1$ means that $A$ is *almost* sure to happen - not that $A$ is sure to happen. Since $P(A|A)=1\ne 0=P(A|A^c)$, the only way to know $A$ surely is to know $A$. – A.S. Jan 24 '16 at 20:54
  • @A.S. what do you mean? Btw how to condition on an event with prob zero...? – BCLC Jan 25 '16 at 19:34
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It holds true practically (almost surely), but not necessarily theoretically (surely). In the original problem the conclusion is only almost surely.

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