Question about sample space

Question

Here, the author says we no longer need to describe $(\Omega,\mathcal{F},\mathbb{P}$). But I wonder what the sample space is. Thank you for your help in advance!

Answer 1

The elements of the sample space fully describe what happened in the experiment. Often, there are a few different choices you could make. Here, I might describe the sample space as follows: Label all the balls: $W^1_1, W^1_2$ are the two white balls initially in urn one, $W^2_1,W^2_2, W^2_3$ are the three white balls initially in urn two. Similarly for blue. Then your sample space consists of pairs $(B_1,B_2),$ where $B_1$ is the ball you transferred from urn one to urn two and $B_2$ is the ball you drew from urn two at the end. Each element in $\Omega$ has the same probability provided it's 'legal' (i.e. $B_1$ is one of the balls in urn one and $B_2$ is either a ball from urn two or it is $B_1$), otherwise has probability zero.