Wait so probabilities of 0 or 1 CAN change? $P(A|B)=1$ does not imply $P(A)=1$ because $0 < P(A=B) < 1$?

Question

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?

Answer 1

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

2. "$P(A|B_1) = 1$ (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.