Relabel one die so that every total is equally likely

Question

You have two fair six-sided dice. The first is an ordinary die with faces 1 to 6. You erase the faces of the second die and write a whole number of your choice (zero allowed, repeats allowed) on each of its six faces.

Relabel the second die so that when both dice are rolled, the total is equally likely to be each of 1, 2, 3, ..., 12. Show that your labelling works and explain why the numbers on the second die are forced.

Accepted Answer

Label the second die 0, 0, 0, 6, 6, 6. Why it works. The first die shows each of 1 to 6 with probability 1/6. The second die shows 0 with probability 1/2 and 6 with probability 1/2, independently. So the total is "the first die's value" half the time and "the first die's value plus 6" the other half. Each of the totals 1 to 6 therefore has probability 1/6 × 1/2 = 1/12, and so does each of 7 to 12. All twelve totals are equally likely. Why the labels are forced. The total 1 must be possible, and it can only arise as 1 + 0, so the second die has a 0. The total 12 must be possible, and it can on…

Relabel one die so that every total is equally likely

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