Reality Is Ugly

"The next thing we decided was that figuring out all the combinations was going to be laborious. We started making some tables but quickly gave up. Monday said he wanted to sleep on it a bit." Roscoe lit up a stogie, and I figured the rest of the tale would come out now.

"Wouldn't you know it, Monday came back with the answer the next day," continued Roscoe.^[6]

^[6] Some of you may have come to the conclusion that Roscoe is an avatar for your humble author. If that is true, then Monday is an avatar for our son, David, who figured out the solution to this problem. I may be good at discovering interesting problems, but David is even better at solving them, and he's an avid baseball aficionado as well. And the name of the avatar is Monday, because that is the day we come to work.

"Monday concluded that the first thing to do was decide how many probabilities were possible as an upper limit. Because there were only 216 different ways to roll the dice, that had to be the maximum. So the best we could possibly do was to cover the interval from zero to 1 in 216 steps. From a granularity point of view, the optimum solution would be to have equal steps of 1/216, or 0.00463."

"Well," I said, "we know we can generate the first one!"

Roscoe grinned.

"Why was it important to try to determine the maximum possible number of probabilities?" I asked.

"Well," said Roscoe, "if you don't do that, you have a problem. Suppose you muck around and find 87 distinct probabilities, up from the 21 we have found so far.^[7] How do you know if you've got them all, and not missed some? Once you know the maximum possible, you can stop if you attain it. If not, you have to figure out why you couldn't get the others.

^[7] As Roscoe himself did in an early attempt to solve this problem.

"Actually, all we have to do is figure out if we can do 108 possibilities up to a probability of 0.5. Then we can get the other 108 by taking the complementary solution," said Roscoe.

That made sense. This is a common "trick" in this domain. If you know that rolling a 3 has a probability of 0.00463, then rolling anything but a 3 will have a probability of (1 - 0.00463), or 0.99537. So getting to a probability of 0.5 is always good enough.