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Nassim Nicholas Taleb says here

no probability that is 0 or 1 should ever change.

Despite these 6 questions

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

  2. https://math.stackexchange.com/questions/1494682/is-it-correct-to-say-that-pa-1-to-pab-1-pa-1-to-pab-pa

  3. https://math.stackexchange.com/questions/1515756/is-a-probability-of-0-or-1-given-information-up-to-time-t-unchanged-by-informati

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

  5. Is $E[1_A | \mathscr{F_t}] = 0 ~\text{or} ~ 1 \ \Rightarrow E[1_A | \mathscr{F_{s}}] = E[1_A | \mathscr{F_t}]$ is only almost surely?

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

It was pointed out to me in maths educator se that...

Actually, $P(A|B)=1$ does not imply $P(A)=1$ because $0 < P(A=B) < 1$.

  1. In fact I believe '$P(A|B)=1$' is equivalent to '$A \subseteq B$ a.s.' i.e. '$A^c \cap B = \emptyset$ a.s.' meaning $P(A^c \cap B)=0$ (where events are almost surely equal or subset/superset if the corresponding indicator random variables are almost surely, resp, $=$ or $\le$/$\ge$). Right?

  2. What's going on? NNT is wrong? Or NNT is right because e is referring to future information and replacing $B$ with $\Omega$ is actually the basic probability version of coarser partitions in filtrations in advanced probability i.e. past info instead of finer partitions in filtrations i.e. future info?

BCLC
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    What *exactly* did NNT say and in what context? I got tired of trying to track through your long chains of links, never to land at a clear quotation, so for anyone without more patience than me, (2) is unintelligible. (1) is incorrect, BTW. Perhaps working with an elementary definition of conditional probability (that is, non measure-theoretic) will help make that clear. – whuber Jan 28 '22 at 17:42
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    Consider choosing a random day of the week in some fashion, with each day being equally likely. Let $A$ = "chose a weekday" and $B$= "chose Wednesday". Clearly $P(A|B) = 1$ but $P(A)<1$. – Glen_b Jan 28 '22 at 17:48
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    You are talking here about whether $P(A|B) =1$ implies $P(A) = 1$, but it seems like the rest of the questions you linked as well as the NNT quote are about whether $P(A) = 1$ implies $P(A|B) = 1$. – Jonny Lomond Jan 28 '22 at 18:10
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    The probability "changing" refers to $P(A)$ "changing" into $P(A|B)$ once $B$ is known to be the case. Seems like you're very confused because you're interpreting it the wrong way round. – Jonny Lomond Jan 28 '22 at 18:18

1 Answers1

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There is some ambiguity in what Nassim Nicholas Taleb (NNT) wrote.

Some chief executive was discussing the certainty of a future event. He said "the probability of [the event] happening is 100% now. But it could change in the future". The error is obvious.

Is the executive saying:

  1. "$P(A) = 1$, but if I get more knowledge (B) then $P(A|B) < 1$".

(this is impossible, and would seem to be the error he is indicating?)

or is the executive saying

  1. "$P(A|B_1) = 1$ (where $B_1$ is what I know about the present). Then it is possible that $P(A|B_2) < 1$ (where $B_2$ is what I know about some future time)"?

(This actually is possible. It also seems to be closer to what most people mean if they know $A$ is true with $100\%$ certainty: they really mean $P(A|B_1) = 1$ is true where $B_1$ is everything I know to be true right now).

It seems like your are interpreting NNT as saying the second thing is in error, and treating this as Gospel.

I am not sure that NNT had anything so precise in mind, but if he did it would seem to be the first interpretation.

I think there are also some philosophical issues with modeling people's belief systems using probability spaces. It might be a reasonable model, but I wouldn't make the mistake of confusing the model with reality. People are much more complex creatures than this.

Steven Gubkin
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