prove occurrences are independently of each other

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

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

Re: prove occurrences are independently of each other

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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

$\displaystyle \begin{array}{*{10}{c}} E&E \\ E&O \\ O&E \\ O&O \end{array}$

You can see the $\displaystyle \mathcal{P}(A)=\mathcal{P}(B)=0.5$, two out of four for each event.

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

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