A probabilityminded despot offers a convicted murderer a fin

A probability-minded despot offers a convicted
murderer a final chance to gain his release. The prisoner
is given twenty chips, ten white and ten black. All twenty
are to be 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. The executioner
will then pick one of the two urns at random and
from that urn, one chip at random. If the chip selected is
white, the prisoner will be set free; if it is black, he “buys
the farm.” Characterize the sample space describing the
prisoner’s possible allocation options. (Intuitively, which
allocation affords the prisoner the greatest chance of
survival?)

Solution

Total chips 20

10 white and 10 black

Two urns will be placed

Prob for any one urn = 0.5

Prob for white = 0.5 if equal white and black are placed in one urn

This only will have max chance for the prisoner to set free or otherwise the one urn if with more black is selected he will not get free

Hence Prob for max gain

= 0.5(0.5) = 0.25 if he placed 5 white and 5 black in each urn.

Sample space consists of

I urn          II urn                 Prob for free

(1 w, 9b) (9w 1b)    0.5(0.1+0.9) = 0.5

(2w 8b) (8w 2b)                  0.5

....

....

(9w 1b) (1w 9b)