TYI 54: A Variant of Elchanan Mossel’s Amazing Dice Paradox

Combinatorics and more 2024-01-17

The following question was inspired by recent comments to the post on Elchanan Mossel’s amazing Dice Paradox.

A fair dice is a dice that when thrown you get each of the six possibilities with probability 1/6. A random dice is such that you get the outcome ‘k‘ with probability $p_k$ where $(p_1,p_2, \dots ,p_6)$ is a vector chosen uniformly at random from the 5-dimensional simplex.

Test your intuition:

You throw a random dice until you get 6. What is the expected number of throws (including the throw giving 6) conditioned on the event that all throws gave even numbers?