An unlucky gamble

Suppose you were offered the chance to play a simple gambling game, in which you are invited to bet on the outcome of a die-roll. There are only two bets allowed. You can wager that the die will land 6, or you can wager that it will land any of 1, 2, 3, 4, or 5. In either case, if your wager is successful, you will win the same prize: one dollar.

So the bets are:

Assume that you know, moreover, that the die has no significant asymmetry in its construction. It does not have a physical bias to one or more sides.

Which bet ought you to take? Assuming you would prefer more money to less, it is obvious that you ought to take the bet on 1–5, rather than the bet on 6.

Now suppose that you really do play this game, and you play it at the same time as a friend. You sensibly choose to bet on 1–5. Your friend, bizarrely, insists that she has a hunch that the die will land 6; so that is the bet she takes. The die lands 6. Your friend wins.