Math Help - prove occurrences are independently of each other

1. prove occurrences are independently of each other

I have problems with this task, I couldnt join the lecture and now i dont get it so every help would be appreciated

Throw two times a fair dice. Prove that these occurrences are pairwise independently of each other but not independently of each other.
A = "number of the first cast of dice is even"
B = "number of the second cast of dice is odd"
C= " both numbers are either even or odd"

Thanks

2. Re: prove occurrences are independently of each other

Originally Posted by suslik
Throw two times a fair die. Prove that these occurrences are pairwise independently of each other but not independently of each other.
A = "number of the first cast of die is even"
B = "number of the second cast of die is odd"
C= " both numbers are either even or odd"
First, it is one die and two dice.
The outcome space is
$\begin{array}{*{10}{c}} E&E \\ E&O \\ O&E \\ O&O \end{array}$
You can see the $\mathcal{P}(A)=\mathcal{P}(B)=0.5$, two out of four for each event.

BUT $\mathcal{P}(AB)=0.25$, an even then an odd.

So $\mathcal{P}(A)\cdot\mathcal{P}(B)=\mathcal{P}(AB)$ showing independence.