Jan 31, 2019

In this post we are going to extend the last die rolling problem. What if the die is imbalanced, and each side shows up with a different probability q_i, for i = 1, 2, 3, 4, 5, 6? Remember that you still want to construct a game where you win with probability of p.

The trick here is to make an imaginary balanced die by observing specific die outcomes. We have in total 6⁶ outcomes for a die, but we only want to choose those unbiased outcome. Then we want to use that imaginary die to play a fair game as before.

To construct the imaginary die, let's do a series of 6 rolls, and record the outcomes as shown in the table. We only consider the sequence where all of the 6 sides show up.

For other outcomes of the original die, let's ignore them (this means only 6! out of 6⁶ real outcomes are considered!). You can easily prove each side of the imaginary die shows up with equal probability. Now you have a balanced imaginary die; use this die to construct your game!