Why does a probability of 0 or 1 remain unchanged with new information, intuitively?

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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 :(

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
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
This is relevant here: How does a Bayesian update his belief when something with probability 0 happened? — kjetil b halvorsen, Jan 28 '22 at 17:10

score 1 · Answer 1 · answered Jan 24 '16 at 05:32

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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.

answered Jan 24 '16 at 05:32

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? – 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

score 1 · Accepted Answer · edited Apr 13 '17 at 12:44

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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.

edited Apr 13 '17 at 12:44

Community

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answered Jan 24 '16 at 14:31

BCLC

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Why does a probability of 0 or 1 remain unchanged with new information, intuitively?

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