The Dice Riddle (and the answer)

First, a recap:

Here is a pair of dice I own. If I roll the dice and read the numbers off the top, is it more likely the numbers add up to be 2, or add up to be 12? Or are the two sums equally likely? Why?

In the course of things, it was brought up:

1. While the mathematical probability of rolling a 2 is 1/36 as is the probability of rolling a 12, the probabilities in the riddle aren’t the same.

2. There’s no trickery like considering a row of spots on the six to be equivalent to a 1, or thinking of a roll of 1 and 2 as 12.

3. The picture is important.

Congratulations to Todd Trimble for cracking the riddle!

If you still want to work on it, stop reading here; otherwise you can find my discussion of the answer below.

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Note the concavity (you can see from the light reflection) of the above dice; by “carving out” the numbers in a standard pair of dice, the dice are no longer fair. As Todd puts it:

…there is overall less mass on the side with a ‘6′ than there is on the side with a ‘1′, which would lend a small bias in favor of the dice landing with the 1’s on the bottom and 6’s on top.

Hence you are more likely to roll 12 than 2.

The effect is noticable after about 500 throws or so (which can happen in the context of a classroom activity that collates the results, so if you’ve ever had a strangely biased experiment, now you know why!)

These dice are quite common (I got the above pair from a Yahtzee set) but they aren’t allowed in casinos.

Here’s a pair of casino dice:

While I wasn’t trying to “make a point” with the riddle, I do think this is a fair demonstration that mathematicians need to be careful when applying what they know to physical reality.