The odd thing about odds

18 April 2013

By John Grochowski

Craps is not a family game in the Grochowski household. I do have a craps layout that I use in seminars, but we don’t sit around the kitchen table and roll them bones unless we’re playing Yahtzee or Monopoly.

But there are plenty of dice in the house, and we picked a pair one night when old friend Jeff was over for dinner. Jeff doesn’t gamble, but his brother Jody is a craps player, and apparently Jody has a theory.

“I guess he makes normal bets most of the time, but I don’t really understand the game,” Jeff said while my wife Marcy nodded in agreement. She doesn’t really get the game either. “But he says he keeps track of how many rolls there have been since the last 12. He says one out of every 36 rolls should be a 12. Is that right?”

That’s true as an average, I told him. There are 36 possible combinations of two six-sided dice, and only one of them totals 12. But that doesn’t mean every set of 36 rolls will include a 12. Some sets will include two, three or even more 12s. Many sets won’t have any 12s at all.

“OK, I can see that,” Jeff said. “But Jody says that if there hasn’t been a 12 in a long time, the odds get better. He said the odds are 35-1 at the start, but if there hasn’t been a 12 in 15 rolls, then the odds are only around 20-1 if one out of 36 is going to be a 12.

“I asked him if that didn’t still mean he was a big underdog, and he said that if he waited long enough, the bet paid bigger than the odds. So he waits until there has been no 12 for 15 rolls, then he starts betting on 12.”

I told Jeff that a one-roll bet on 12 pays 30-1 odds. If indeed the odds against rolling a 12 were lowered to 20-1, then making the bet only in those situations would give the player a big edge. But that’s not how it works.

Jeff gave a short laugh and said, “I knew you were going to say that. So tell me, what’s the flaw?”

The flaw is that the odds don’t change, no matter how long it’s been since the last 12. There’s a 1 in 36 change of a 12 on every roll. If there’s been no 12 in the last 50 rolls, the chances on the next roll are still 1 in 36. If there have just been three 12s in a row, the chances of another are 1 in 36.

In the long run, 12-less streaks and 12s bunched together just fade into statistical insignificance.

Then Jeff did something he’s never done before. He asked for a demonstration.

“You have dice, right? Let’s try it.”

I had some misgivings. This was going to be very unscientific and nowhere near long enough to prove any points. We weren’t going to get the same kind of bounce on my kitchen table as on a casino craps table. The table’s not long enough to give anything like a normal craps shooter’s roll. And in the short term, anything can happen, including winning streaks for Jody’s system.

But I went to the desk in my home office and pulled out two sharp-cornered red casino dice, and we went to work. Jeff did the rolling, I marked the results.

The first 12 showed up on the fifth roll. Jeff rolled another 12 just eight rolls later. Jody’s system wouldn’t have called for bet yet, and we started counting again.

Fifteen rolls later with no 12, Jeff called out, “Now’s when he would start betting.”

Then he rolled 6, 9, 8, 11, 6, 8. I told him that if he were on the pass line, he’d have made a couple of points. “I have no idea what that means,” he said.

The no-12 streak grew. 30 rolls in row grew to 40, 50, 60.

“This is incredible,” Jeff said.

Finally at roll No. 63 since the last 12, the double-6s came up once more.

“That’s enough,” Jeff said. “I just wanted to see how it worked.”

Proof? No, just one random streak. But over hundreds and thousands of random streaks, the math of the game will hold up. You can’t get an edge by waiting for the odds to change. They don’t change.

Look for John Grochowski on Facebook (http://tinyurl.com/7lzdt44) and Twitter (@GrochowskiJ).