Probability Space of Tossing Two Dice 
This is one of my homework problems that I can't seem to grasp my head around. I already derived the answer to 3a. However when it comes to 3b and 3c I get lost. Could someone please explain how to go about doing this.
 A: Hints on b):
1) $P_A(B)$ is (probably, check your book on this) the notation for the probability that event $B$ occurs under the condition that event $A$ occurs. Mostly it is denoted as $P(B|A)$ and essential is the equation: $$P(A|B)P(B)=P(A\cap B)$$
2) Two events $U$ and $V$ are independent if $P(U\cap V)=P(U)P(V)$ which is equivalent with $P(U|V)=P(U)$ if $P(V)>0$. 

example on b):
Taking the red as first die and the blue as second we have: 
$A=\left\{ \left(r,b\right)\in S\mid r\text{ odd}\right\} $ and
$B=\left\{ \left(r,b\right)\in S\mid b\text{ even}\right\} $. 
Here
$S=\left\{ \left(r,b\right)\mid r,b\in\left\{ 1,2,3,4,5,6\right\} \right\} $.
Since all outcomes $\left(r,b\right)\in S$ have the same probability
we have $P\left(A\right)=\frac{\left|A\right|}{\left|S\right|}=\frac{18}{36}=\frac{1}{2}$
and $P\left(B\right)=\frac{\left|B\right|}{\left|S\right|}=\frac{18}{36}=\frac{1}{2}$.
Note that $A\cap B=\left\{ \left(r,b\right)\in S\mid r\text{ odd}\wedge b\text{ even}\right\} $
and $P\left(A\cap B\right)=\frac{\left|A\cap B\right|}{\left|S\right|}=\frac{9}{36}=\frac{1}{4}$.
Then $P_{A}\left(B\right)=\frac{P\left(A\cap B\right)}{P\left(B\right)}=\frac{\frac{1}{4}}{\frac{1}{2}}=\frac{1}{2}$.
Then $P_{A}\left(B\right)=P\left(B\right)$ so we conclude that
$A$ and $B$ are independent.
A: Well, I suggest you read a book.
PA(B) is the same as P(B|A), ie, the probability of event B given that event A happens. You have to see what is the formula to solve P(B|A) (see my book suggestion below, where you can find this formula). Alternatively, you can draw a matrix with all possible outcomes (all possible values of the two dices, there should be 36). Then mark all values where the condition must be satisfied (ie, all values where red die is 1,3 or 5) and then all values wherein the marked one where the blue die is 2, 4 or 6. That is the probability.
I liked The Cartoon Guide to Statistics by Larry Gonick and Woolcott Smith. Chapter 3 is what you are looking for. First time I read the book was clear but required to pay attention to the content. Second time was very easily understandable.
