Lessons Learned

Roscoe seemed less than totally happy at the end of the story. "What's bugging you, Roscoe?" I ventured.

"Well, first of all," he replied, "I got suckered once again. I thought I had a baby problem on my hands, and it turned out to be much more complex than I first thought. That's always annoying.

"Second, one of my time-honored techniques washed out on me," he continued. "Usually when a problem starts to grow teeth, I revert to a simpler instance of the problem to gain insight. But in this case, the simpler case of the sum of two dice was totally useless.

"Third, Monday's solution is convincing, especially when you finally have the table in hand. But even there, it seems like he used some heuristics instead of a logical proof. Although I am the last guy in the world to criticize anyone for getting the answer any way he can. Never let it be said that I was an advocate of excess purity, let alone elegance."

Now, I take a much more clement approach. Roscoe, and especially his buddy Monday, had done a great job using pencil, paper, slide rule, and common sense. In fact, had Roscoe lost his slide rule in the shipwreck, Monday still could have solved the problem, as the division by 216 is a purely cosmetic eventwe can state probabilities as 67/216 if we want to.^[8] That the problem can be solved on a desert island, with no computers, no Microsoft Excel, and no pivot tables by using just time, energy, and intellectual curiosity, is wonderful. It also means that Blaise Pascal could have solved the problem in the 17th century; he had all the tools he needed back then. And he didn't lack for intelligence or intellectual curiosity, either. Ironically, Hamming notes that sometime before the year 1642 Galileo was asked about the ratio of the probabilities of three dice having either a sum of 9 or else a sum of 10. He got at least as far as constructing our Table 20.1, correctly deducing that rolling a 10 was slightly more probable than rolling a 9. So people were thinking about these things long before baseball.

^[8] Or Roscoe could do long division by hand if he had to.

Roscoe and Monday wound up being able to extract a uniform probability distribution with a granularity of about half a percent ^[9] by simply rolling three dice and lookingat the total, so long as they used the algorithm of specifying the probability first and then deciding whether the roll was successful or not. In the original baseball example, it meant they could get batting averages to within five points.^[10]

^[9] One part in 216 is slightly better than half a percent; one part in 200 would be exactly half a percent.

^[10] The late Ted Williams hit .406 in 1941 and is the last player to have batted over .400 for a season. Hence, Table 20.6 is more than adequate for batting averages. For other probabilities greater than 0.500, one needs to extend the table symmetrically, using the complementarity principle mentioned earlier.