Probability - Forming an equation

[mattgad]

Two events R and S are independent. P(R|S) = 3/4 and P(S) = P(S' n R'). By letting x equal P(S) and forming an equation for x, or otherwise, find
(a) P(S)
(b) P(S' n R)
(c) Write down P(S|R).

How would I go about doing this?

Thanks,

 

[pka]

If A and B are Independent, Pr(A|B) = Pr(A)!

Moreover A, B, A’ and B’ are all pairwise independent!

Therefore, P(R)=3/4 and P(S’R’)=P(S’)P(R’)=P(S).

 

[mattgad]

I'm sorry but I don't see how I can use this data to find P(S). Am I missing something?

 

[pka]

Because S’ and R’ are independent, we have
\(\displaystyle P(S' \cap R') = P(S')P(R') = P(S)\).

So \(\displaystyle \L
P(S')P(R') = \left( {1 - P(S)} \right)\left( {1/4} \right) = P(S)\).

Solving we get \(\displaystyle \L
5P(S) = 1\quad \Rightarrow \quad P(S) = 1/5\)

 

[mattgad]

Thanks for your help. I understand and managed to complete the rest of the question.