I have this maths problem that I'm trying to figure out. It looks really simple but it confuses the heck out of me.
The question: Let $X$ and $Y$ be two independent random variables so that $X \sim Bernoulli(1/2)$ and $Y \sim Bernoulli(1/4)$. Find $P(X=1 | X=1 \text{ or } Y=1)$.
If $X$ and $Y$ are independent, does it really matter what $Y$ is?? I have a formula for calculating conditional probability of joint distributions involving two discrete random variables, but I don't understand how to apply it in this case. Or if it should even be applied in this case. I think it's the "or" that trips me up.
$$P(X=1|X=1 \,\,\text {OR} \,\,Y=1)= P(X=1|(X=1)\cup (Y=1))$$ $$=\frac {P(X=1)} {P(X=1)\cup (Y=1)}=\frac {P(X=1)} {P(X=1)+P(Y=1)-P(X=1, Y=1)}$$ $$=\frac {P(X=1)} {P(X=1)+P(Y=1)-P(X=1) P(Y=1)}.$$
I have use the following facts:
$A \cap (A\cup B)=A$
$P(A\cup B)=P(A)+P(B)-P(A\cap B)$