Distributions from selections and stopping rules

Selection questions split into two forms: a fixed-size group, or an ordered process that continues until something happens.

If 2 counters are selected from 3 red and 5 blue counters and $R$ is the number of red counters, then

Use $r=0,1,2$ to produce the three table cells.

The denominator counts all two-counter groups. The numerator chooses exactly $r$ red counters and the remaining $2-r$ blue counters. Every table cell must use the same sample space.

Without replacement means a selected object is not returned, so the numbers and probabilities change after every draw. If $X$ is the draw on which the first green counter appears, then $X=k$ means: non-green on the first $k-1$ draws, then green.