let $S=f(Y,Z)$, where $Y$ and $Z$ are two independent random variables and where $f$ is a specific function; $f(Y,Z)=S=YZ(Y+Z+1)^{-1}$. Let $\hat{S}=f(Y=c,Z)$, in which $Y$ is replaced by a (positive) constant $c$; $\hat{S}=cZ(c+Z+1)^{-1}$. We define $a$ as a positive constant.
I am trying to compute the probability of the following event:
$S \ge a$ and ($\hat{S} < a$ or ($\hat{S} \ge a$ and $Y < c $)).
Question: How can I write this event in function of the probabilities of the random variables defined above ?
First step: P($S \ge a$) P($\hat{S} < a$ or ($\hat{S} \ge a$ and $Y < c$)). Is this first step correct ? I am confused because $S$ and $\hat{S}$ both depends on $Z$. Also, how can I continue ?
Edit: As pointed out by @Did, the first step I have written is not correct since the two events, i.e. $A=(S \ge a)$ and $B=(\hat{S} < a$ or$(\hat{S} \ge a$ and $Y < c))$, are not independent.
Let us define $B_1=(\hat{S} < a)$ and $B_2=(\hat{S} \ge a$ and $Y < c)$.
Hence, $P(A \cap B)=P(A \cap (B_1 \cup B_2))=P((A \cap B_1) \cup (A \cap B_2))$.
1- We know that $P((A \cap B_1) \cup (A \cap B_2))=P(A \cap B_1)+P(A \cap B_2)-P((A \cap B_1) \cap (A \cap B_2))$.
If I am not mistaken, we have $P((A \cap B_1) \cap (A \cap B_2))=0$ since $(A \cap B_1)$ and $(A \cap B_2)$ are disjoint (because $B_1$ contains $\hat{S}\ge a$ whereas $B_2$ contains $\hat{S}<a$). Am I correct ?
2- $P((A \cap B_1)= P(S \ge a \cap \hat{S} <a)=P(S \ge a \mid \hat{S}<a)P(\hat{S}<a)$.
3- $P((A \cap B_2)= P(S \ge a \cap (\hat{S} \ge a, Y < c))=P(S \ge a \mid \hat{S}\ge a, Y <c)P(\hat{S}\ge a, Y <c)$.
Could someone please tell me if this approach is correct ? is there any other (easier) approach that I can adopt ?