First Steps

"Monday got right on the basic combinatorics. With three six-sided dice, there were 6 x 6 x 6 possible outcomes, or 216 possibilities, but only 16 different totals. So Monday constructed Table 20.1.

Table 20.1. Ways to Form Various Totals

Total	Number of Ways to Roll that Total
3	1
4	3
5	6
6	10
7	15
8	21
9	25
10	27
11	27
12	25
13	21
14	15
15	10
16	6
17	3
18	1
Sum	216

"Looks good to me," I replied. "There is some obvious symmetry. For example, both 3 and 18 come up exactly once, as you would expect. And 4 and 17 are the same, and so on. The most frequent occurrences are 10 and 11, as there are lots of combinations that will yield those totals. And the number of total ways adds up to 216, so you can't be too far off the mark. But it looks like there are, so far, only eight distinct probabilities in the offing."

"Appearances can be deceiving," said Roscoe. "But let's add in the probabilities we have so far to double-check our work." He then produced Table 20.2.^[5]

^[5] Note that the probabilities have, in some cases, five significant figures. On his desert island, Roscoe and his slide rule could do three at best. 0.00463 represents three significant figures; the others should be appropriately rounded. Even though the decimal probability is infinitely precise, being the quotient of two integers, Roscoe can only achieve three places with his slide rule; on the other hand, if he does long division by hand, he can get as many as he has the time and patience for.

Table 20.2. Probabilities Corresponding to Each Total

Total "N"	Number of Ways to Roll that Total	Probability of "N"
3	1	0.00463
4	3	0.01389
5	6	0.02778
6	10	0.04630
7	15	0.06944
8	21	0.09722
9	25	0.11574
10	27	0.12500
11	27	0.12500
12	25	0.11574
13	21	0.09722
14	15	0.06944
15	10	0.04630
16	6	0.02778
17	3	0.01389
18	1	0.00463
Sum	216	1.00000

"Whoop-de-do!" I exclaimed. "Roscoe can divide by 216."

"Calm down, Sonny," said Roscoe. "The fun is only beginning."