A simple question I tried to ask in a little different form on another site, evidently without much success in making myself clear. I am retrying with (hopefully) clearer language.
I was looking at the definition of independence for two events. $P(A|B) = P(A)$, with $P(B)$ not zero.
My little doubt is about a specific case where $A$ is strictly included in $B$. That is, $A$ implies $B$.
In this case, $P(A|B) = P(A) / P(B)$. Now assume that $P(B) = 1$.
In this case, $P(A|B) = P(A$) which satisfy the independence definition (does it?)
How is this reconciled with the fact that A implies B (and therefore cannot be "independent")? [to say that the occurrence of A guarantees the occurrence of B because it is a part of it seems to be the conceptual opposite of "independence".]
What am I getting wrong?
update
Link to the question mentioned in the comments. Following a request by David K previous question mentioned in comments
Preliminary comments
Mathematical definitions are, by definition, mathematical. While we would like them to align with our preconceived notions of meaning when they use common words (such as independence), sometimes the best we can do is to get them to line up in the "normal use case," rather than in every possible "edge case" where the definition might apply.
Even the word "implies" in "$A$ implies $B$" has a special definition in mathematics that many people find unintuitive. In particular, many people have trouble accepting the notion that a false statement implies all other statements, regardless of the truth of the other statement or the ability to derive one statement from the other. In terms of sets, the empty set is a subset of every set. (Many people have trouble with this fact too.)
People also have trouble understanding why "$A$ implies $B$" must be true when $B$ is true, regardless of the truth of $A$ or the ability to derive one statement from the other.
For that matter, many people balk simply at the inability to derive one statement from the other, for example, "If Tierra del Fuego is in the southern hemisphere then Henry VIII of England had six wives." Mathematically, that's a true implication, but how are the number of Henry's marriages and the location of Tierra del Fuego not independent of each other?
The case where $P(B) = 1$ is definitely an "edge case." You are apparently unconcerned by the inability to use the formula $P(A) = P(A\mid B)$ in the case where $P(B) = 0$, so ask yourself why you are concerned with the interpretation of the formula for the case where $P(B) = 1$, and whether that concern is really relevant to your ability to use this definition.
I think a much bigger objection to this definition is that it allows $A$ to be independent of $B$ while $B$ is not independent of $A$ (in particular, when $P(A)=0$). That asymmetry bothers me more than dealing with the edge cases. I prefer the definition that events $A$ and $B$ are independent when
$$ P(A \cap B) = P(A) P(B). $$
Not only is this symmetric in $A$ and $B$, it also makes the edge cases $P(B)=0$ and $P(B)=1$ both relevant to the definition.
Discussion
Independence of random events is defined so that it is about probability. Logical deduction is not the goal.
In your example, where $P(B)=1$, if you find out that $A$ occurred, what does it tell you about the probability of $B$?
It tells you nothing. The probability of $B$ given $A$ must be $1$ since $A$ implies $B$, but you already knew that the probability of $B$ was $1$ before you had any information about whether $A$ occurred.
Likewise, the occurrence of $B$ doesn't tell you anything new about the probability of whether $A$ occurred.
The most obvious case where $P(B)=1$ is the case where $B = \Omega$, that is, $B$ is the entire probability space. What sense does it make to say that there is a dependence in probability between the entire probability space and some other event? Since the event $\Omega$ is absolutely guaranteed to occur no matter what, in what way can it depend on some other event, or some other event depend on it?
Consider in particular the case where $A = \emptyset$, that is, $A$ is the null event, and therefore $P(A) = 0$. This is the one event that absolutely cannot occur under any circumstances. And it is certainly true that $\emptyset \subseteq \Omega$. How does this link between $\emptyset$ and $\Omega$ prevent us from saying that these two events are independent?
The cases where "independent events" trouble you are cases where the word "implies" (or the words "if" and "then") trouble many people. I will venture to say that the problem is not just with the definition of "independence", but also with the definition of "implies".
Finally, note that in every case where $P(A) \neq 0$ and $P(B) \neq 1$ -- that is, in every "ordinary" case -- $A \subseteq B$ implies that $A$ and $B$ are not independent events. This aligns well with intuition. It's just in some weird edge cases that we get weird results from the definition, and patching the definition to eliminate those results would just make the definition more complicated and harder to use in practice.