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

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 where 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?

Addition: Test your intuition also on the following simpler question. We have a coin that gives head with probability p and tail with probability 1-p, and p is itself chosen (once and for all) uniformly at random in [0,1]. What is the expected number of coin tosses until we get head.

Addition:  Note that there are two different ways to understand the question itself. One way is to regard the conditioning as applying both to the dice throwing and to the random choice of the probabilities . A second way is to consider the parameters as being determined at random uniformly; and then taking the average over for the expected number of throws conditioned on the event that all throws gave even numbers.