Terence Coelho

Which of the following do you think is bigger? A: The expected number of rolls of a fair die until you roll two 6s in a row, given that all rolls were even. B: The expected number of rolls of a fair die until you roll the second 6 (not necessarily in a row), given that all rolls were even. If you are unfamiliar with conditional expectation, think of it this way: Imagine you were to perform a million sequences of die rolls, stopping each sequence when you roll two 6s in a row. Then you throw out all the sequences that contain an odd roll. The average number of rolls in the remaining sequences should be close to A. Next, perform a million sequences of die rolls, stopping each sequence when you roll the second 6. Throw out all the sequences among these that contain an odd roll. The average number of rolls in the remaining sequences should be close to B. I asked something like this on r/math about a year ago, and even with the hint that the answer was paradoxical, the early consensus was that A must be larger. The justification was more or less the following: any time you roll until reaching two 6s in a row, you will have also hit your second 6 at or before then. So regardless what the conditions are, A must be larger than B. But the correct answer is actually B. What on earth is going on? A quick verification Before we proceed, let's write some code to estimate A and B. The only goal here is to be as unambiguous as possible, so the code will be almost comically unoptimized in both run-time and length. import random def estimate_A(n): #Rolls 'n' sequences of die rolls, stopping each when two 6s in a row. #Tracks number of sequences with no odds #Tracks sum of number of rolls in all sequences with no odds #Computes average by dividing the two. num_sequences_without_odds=0 sum_rolls_without_odds=0 for i in range(n): counter=0 last_roll_six=False has_odd=False #Roll until two sixes in a row