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

no probability that is 0 or 1 should ever change.

Despite these 6 questions

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?

• 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, 2022 at 17:42
• 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 $P(A)<1$. Jan 28, 2022 at 17:48
• 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 $P(A|B) = 1$. Jan 28, 2022 at 18:10
• 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. Jan 28, 2022 at 18:18

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.