A soft question about probabilistic inference

Question

Suppose A and B are two propositions and $ A \implies B$ is true.WE know all it means is that B is true whenever A is true. Now consider a situation where we only that A is true with probability ,say,$p.$ The thing I am thinking about is what we can say about $P(B).$ I think we can only legitimately conclude that B is true with probability at least $p$ and not that $P(B)=p$.Is my intuition right here?I do not know how to justify it.However I made the following example to illustrate it .
Define A nd B as:
A:a number is divisible by 10
B: a number is divisible by 5
Then clearly $A \implies B$.
Now consider the experiment of choosing a number randomly from the set:$$ S=\{5,10,15,25,35\}$$ Obviously,$A \implies B $ is true here also but $P(A)=1/5$ and $P(B)=4/5 \neq1/5$.My related question is the following:

QNo2:Suppose a statement P holds with probability at least $a(n),n \in \mathcal{N}$ where $a(n) \rightarrow 0$ as n goes to infinity.How do we interpret that in common sense terms?

Thank you for any clarifications/responses

Answer 1

Let $\Omega$ be the probability space. In your example, $\Omega=S$. We consider $A$ and $B$ are two events. Since $A\Rightarrow B$, i.e., $A$ occurs then $B$ occurs, we have that $A\subset B$. Therefore, $P(A)\leq P(B)$.

P/S: Without the context, it is difficult to interpret your QNo2.