Link between P(A) and P(A|B) I feel like there's no link between P(A) and P(A|B), where A and B are random events. For example, P(A) can be 0.3 and P(A|B) can be 0.9.
However, I'm not so sure of this when P(A) is either 0 or 1.
I mean:


*

*P(A)=0 => P(A|B)=0, for every random event B ?

*P(A)=1 => P(A|B)=1, for every random event B ?

 A: You have issues when $\mathbb P (B)=0$ but is not impossible  
Consider for example $X$ a random variable with a standard Gaussian distribution $\mathcal N(0,1)$


*

*Let $A$ be the event $X^2=4$ and $B$ the event $X=-2$.  Then  $\mathbb P (A)=0$ but $\mathbb P (A\mid B)=1$

*Let $A$ be the event $X^2 \not = 4$ and $B$ the event $X=-2$.  Then  $\mathbb P (A)=1$ but $\mathbb P (A\mid B)=0$
If you know $\mathbb P (B)$, then there are bounds relating $\mathbb P (A)$ and $\mathbb P (A\mid B)$: $$\mathbb P (A\mid B){\mathbb P (B)} \le \mathbb P (A) \le \mathbb P (A\mid B)\mathbb P (B)+(1-\mathbb P (B))$$ which if $\mathbb P (B)$ is non zero is equivalent to $$1-\dfrac{1-\mathbb P (A)}{\mathbb P (B)} \le \mathbb P (A\mid B) \le  \dfrac{\mathbb P (A)}{\mathbb P (B)}$$
A: P(A) and P(A|B) are linked, since the occurrence of B does affect the probability of the occurrence of A.
example:


*

*P(A) = You pass your exam = 0.3

*P(B) = You study for you exam

*P(A|B) = You studied, so there is a higher chance of passing your exam = 0.9


However, point 1 and 2 are true:
1) P(A) = 0, event that will not occur. No prior event can change that.
2) P(A) = 1, event that will definitely occur. No prior event can change that.
