Different people use different types of logic. I try to use “deduction,” whereby I learn the general rule and then apply that to specific situations. A friend of mine, Abe, uses “induction,” which is to look at specific events and then form a general rule that these events will usually reoccur in similar situations.
This difference in reasoning came about when I was trying to make a point to Abe about dealt full houses. I mentioned that dealt full houses occurred about once an hour. He immediately challenged me. “Just yesterday,” he said, “I played nine hours at the Palms on 25¢ Ten Play Jacks or Better and only received two dealt full houses. Your rule must be wrong.”
I went through the math with him. Every full house contains a 3-of-a-kind and a pair. There are 13 different ranks for the trips and once you have one of those ranks selected, 12 different ranks for the pair. (Or alternatively, 13 different ranks for the pair and once you have the pair, 12 different ranks for the trips.)
There are four ways to get each 3-of-a-kind. Using Kings as an example and small letters to stand for clubs, diamonds, hearts, and spades, we can have Kc-Kd-Kh; Kc-Kd-Ks; Kc-Kh-Ks; and Kd-Kh-Ks. An alternative way to look at this is that the first group has all the Kings except the Ks; the second one has all the Kings except the Kh; the third one has all the Kings except the Kd; and the fourth one has all the Kings except the Kc. Then there are six ways to get a pair. Using sixes as an example, we have 6c-6d; 6c-6h; 6c-6s; 6d-6h; 6d-6s; 6h-6s. Order doesn’t matter in these situations, so Kc-Kd-Kh is exactly the same as Kc-Kh-Kd and Kd-Kc-Kh and Kd-Kh-Kc and Kh-Kc-Kd and Kh-Kd-Kc.
The appropriate mathematical procedure is to multiply these numbers: 13 * 12 * 4 * 6 = 3744. Since there are 2,598,960 different starting hands in a 52-card deck, dividing the first by the second gives us a quotient of 694.1667 for a dealt house frequency of once every 694.1667 deals. If you play 700 hands per hour, on average you’ll get slightly more than one dealt full house per hour.
To me, this ended the debate and I could then go on to talk about the point I was trying to make. Not to Abe. He couldn’t get around the fact that he recently played nine hours and only got two dealt full houses. I thought that fact was irrelevant. In some nine-hour sessions, he’ll get 15 full houses. These things vary. I was talking about an average and Abe couldn’t get past the specific.
On another occasion, Abe and I were discussing a play at craps. For reasons I’ll go into some other day, I suggested that he play one hand of $1,000 every month. His expected loss would be $14 each month since craps has a 1.4% house edge on either the Pass or the Don’t Pass bets. Over a year, he would, on average, lose $158 (i.e. 12 * $14). The actual distribution may be calculated by using the binomial theorem:
| Lose $12,000 | 1-in-3467 |
| Lose $10,000 | 1-in-297 |
| Lose $8,000 | 1-in-56 |
| Lose $6,000 | 1-in-17 |
| Lose $4,000 | 1-in-7.8 |
| Lose $2,000 | 1-in-5.0 |
| Break Even | 1-in 4.4 |
| Win $2,000 | 1-in-5.3 |
| Win $4,000 | 1-in-8.8 |
| Win $6,000 | 1-in-20 |
| Win $8,000 | 1-in 70 |
| Win $10,000 | 1-in-393 |
| Win $12,000 | 1-in-4851 |
Making this bet in a particular way should gain Abe benefits of perhaps $1,000 or so per year while losing an average of $158 (Again— details later. Today’s column is about something else.)
I spread out all of this deductive logic before Abe, but he couldn’t get by the chance that he could lose $10,000 or $12,000 a year and that the benefits might be nowhere near $1,000. Abe remembers a time when he lost 12 hands in a row at craps and is fearful that it might happen again. Abe is correct in that it COULD happen. But the odds are 3466/3467 against that happening over the next twelve decisions. Abe’s memory of that happening makes it seem to him almost 50-50 that it will happen again this year.
This deduction-versus-induction debate isn’t about who’s smarter. It’s about whose way of thinking is more applicable to winning at gambling. If you’re going to be successful at figuring out how to beat promotions, you’re probably going to have to use deductive logic. The way Abe uses inductive logic is that he remembers bad streaks of all sorts that have happened to him and thinks they are more common than they really are. He’s probably has averaged being slightly ahead over the past several years, with some years up and some years down. When he’s winning, he tends to be optimistic about what can happen. When he’s losing, be becomes very pessimistic. Since he’s losing so far this year, he’s very pessimistic and it’s tough to convince him of the benefits of any play.
Abe believes that if it happened once, it is very likely to happen again. I believe that even though some rare occurrence happened before, it’s still a rare occurrence. I’ll give a weight to it (perhaps 1-in-10; 1-in-100; 1-in-10,000, etc. — whatever is appropriate) but I won’t let the fact that it happened recently trick me into believing that it’s 50-50 and it will soon happen again.