This has the strong whiff of Monty Hall. But is not the same because Monty cannot choose the door to reveal to you.

Do members know of this paradox? If so, can they enlighten me as to how the rather daft argument that you should toss a coin is wrong? Otherwise, I will have another sleepless night!

One argument that occurs to me is that if you were told what round you were in, then the justification for the coin toss evaporates. You are not 1 member of an increasing unknown sequence. You will take the roll of the dice.

The round you get to is completely random and a appears irrelevant to decision making. My PhD was on conditional inference and so-called ancillary statistics and the number of rolls of a die without hitting 6 looks pretty damned ancillary to me! But the catch is that it is unobserved so we cannot condition on it.