Generating More Probabilities

"Hmm," I said. "Let's add a column to your table. We can generate some new probabilities by considering more 'cumulative' outcomes." So I augmented Roscoe's table to produce Table 20.3.

Table 20.3. Augmented Probability Table

Total "N"	Number of Ways to Roll that Total	Probability of "N"	Probability of "N or Less"
3	1	0.00463	0.00463
4	3	0.01389	0.01852
5	6	0.02778	0.04630
6	10	0.04630	0.09259
7	15	0.06944	0.16204
8	21	0.09722	0.25926
9	25	0.11574	0.37500
10	27	0.12500	0.50000
11	27	0.12500	0.62500
12	25	0.11574	0.74074
13	21	0.09722	0.83796
14	15	0.06944	0.90741
15	10	0.04630	0.95370
16	6	0.02778	0.98148
17	3	0.01389	0.99537
18	1	0.00463	1.00000
Sum	216	1.00000

"Well, you're on the right track," Roscoe offered. "Your fourth column definitely adds some new probabilities. For example, the probability of shooting either a 3 or a 4 is 0.00463 + 0.01389, which you have calculated to be 0.01852. You express this '3 or 4' as the probability of '4 or less.' By '5 or less' you mean that rolling a total of either 3, 4, or 5 defines a successful outcome.

"In fact, that's exactly how Monday and I proceeded. Problem is, you haven't gone nearly far enough."

"Before you jump ahead," I said, "let's look at how many probabilities we have so far. I'll shade the non-redundant ones in the table." I added some shading to produce Table 20.4.

Table 20.4. Augmented Probability Table with Shading

Total "N"	Number of Ways to Roll that Total	Probability of "N"	Probability of "N or Less"
3	1	0.00463	0.00463
4	3	0.01389	0.01852
5	6	0.02778	0.04630
6	10	0.04630	0.09259
7	15	0.06944	0.16204
8	21	0.09722	0.25926
9	25	0.11574	0.37500
10	27	0.12500	0.50000
11	27	0.12500	0.62500
12	25	0.11574	0.74074
13	21	0.09722	0.83796
14	15	0.06944	0.90741
15	10	0.04630	0.95370
16	6	0.02778	0.98148
17	3	0.01389	0.99537
18	1	0.00463	1.00000
Sum	216	1.00000

"So I count 8 in the first column and 14 in the second column, for a total of 22. Is that what you get?" I asked.

"Not exactly," said Roscoe. "We have what might be called an accidental degeneracy. Note that the probability of rolling a 6 is identical to the probability of rolling a '5 or less.' That's because there are 10 ways to make a 6, and (1 + 3 + 6 = 10) ways to make a 3 or 4 or 5, which is 5 or less. So I guess you have 21 distinct probabilities so far.

"But, as I said, you haven't gone far enough. There are lots more combinations."