I've been working through a text on probability and have been having trouble understanding how to approach this question.
The premise is that you toss a coin. If Heads, you pick a ball from Urn 1. If Tails, you pick a ball from Urn 2.
Urn 1 has 3 red balls, 3 green balls. Urn 2 has 4 red balls, 2 green balls.
You then draw two balls from the selected urn, with replacement.
Let $R_1$ be the event that ball 1 is red, and $R_2$ the event that ball 2 is red.
By the law of total probability, I can see that P($R_1$) = P(R|H)P(H) + P(R|T)P(T), and the same procedure can be followed for P($R_2$), giving $\frac{7}{12}$.
However, the text then asks me to find $P(R_2|R_1)$ WITHOUT assuming independence. (The intention is to then prove/disprove independence). I can't see how to do this at all. Any guidance is appreciated!
The phrase "without assuming independence" is a redundant distraction. As Michael says in his comment, just use the standard formua $$ P\left(R_1|R_2\right)=\frac{P\left(R_1\cap R_2\right)}{P\left(R_2\right)}\ . $$ If this turns out to be $\ \frac{7}{12}\ $, the same as the unconditional probability, $\ P\left(R_1\right)\ $, then the events $\ R_1\ $ and $\ R_2\ $ are independent, otherwise they're not.