"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.
"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.
"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."