Probability Crosstabulation Method

Mesmer
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I don't really understand the tabulation method of computing probabilites that my professor has introduced us to, so I'm trying to get a little help on the following problem. I don' t know how to code an array or table in latex so to make my post look better I used an attachment. and use imageshack to host the image I'm really sorry about that.

http://img407.imageshack.us/img407/2716/probabilitynd3.jpg"
I really don't know where to begin I just need a push or two in the right direction. My probability book does not cover this method so all I have for a reference is my class notes
 

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Is no one familiar with this method? If so this bodes ill for this semester :(
 
Can someone just point me to a webpage or other source that has a few examples of this kind of crosstabulation?
 

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...
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