Answer
The sample space describing the prisoner’s possible allocation options are:
\[S=\{{{W}_{1}}{{U}_{1}},\text{ }{{W}_{9}}{{U}_{2}},\text{ }{{B}_{10}}{{U}_{2}}\}\]
Work Step by Step
There are 20 chips, 10 whites and 10 black which are placed into two urns. According to any allocation scheme the prisoner wishes, with the one proviso being that each urn contain at least one chip. He will pick one of the two urns at random and from that urn, one chip at random.
Let ${{U}_{1}}$, ${{U}_{2}}$ denote urn 1 and 2 respectively and $W$, $B$ denote white and black chips, respectively. There is a ½ chance of choosing any urn.
Assume that the executioner will then pick ${{U}_{1}}$ at random that contains one white chip, so urn ${{U}_{2}}$ contains 9 white and 10 black chips. Hence, this allocation will have the maximum chance for prisoner survival.
Therefore the sample space can be written as:
\[S=\{{{W}_{1}}{{U}_{1}},\text{ }{{W}_{9}}{{U}_{2}},\text{ }{{B}_{10}}{{U}_{2}}\}\]
