>>12095000

You need to figure out how many rolls yield a given value [math]k[/math] and divide this by the total number of possible rolls ([math]400[/math] in this case, which I hope is obvious).

When trying to calculate the number of rolls which output a given value, start small. Clearly, [math](1,1)[/math] is the only roll which will yield [math]1[/math]. Then [math](1,2)[/math], [math](2,1)[/math] and [math](2,2)[/math] will output [math]2[/math]. If you imagine the problem visually (as a [math]20 × 20[/math] grid), a pattern is already forming. [math]k=1[/math] gave a [math]1 × 1[/math] square, then [math]k=2[/math] extended this to a [math]2 × 2[/math] square. As we increase [math]k[/math] we just keep adding L-shapes to extend this square. For any value of [math]k[/math], this L-shape consists of [math]2k - 1[/math] outcomes (I'll let you verify this yourself). This gives us a probability distribution defined by [eqn]P(Y = k) = \frac{2k - 1}{400}[/eqn]