A Certain Kind of Approach

A month or two ago, I mentioned on the Gambling with an Edge podcast that I have a buddy with two kids — “Jack,” 12 years old and “Mary Ann,” 10 years old — who are becoming fascinated with the game of backgammon. I’ve agreed to provide them with some backgammon instructions, and I’m enjoying the process of teaching them. I’ve taught adults for years but have had limited experience with teaching children.

The lesson on this particular day was about the doubling cube.

“Let’s say,” I began, “that from a certain position, your opponent will win 26 times out of 36 and you will win 10 times out of 36.” Backgammon players will have no trouble constructing one or more positions that meet this criterion, but I want today’s column to be understood by those readers who do not understand backgammon as well as those who do.

“Let’s say that you are playing for $1 and your opponent,” I continued, “offers you the doubling cube.  What this means is that you have the choice of accepting the cube and playing out the game for $2 or passing the cube and conceding $1 right now. What would you do?”

Both kids are pretty bright and are in STEM schools, which specialize in science and math, but the boy is two years older.  When it comes to figuring out mathematical problems (which is what I thought this was), those extra two years make a difference.

At this point in time, neither knew how to figure this out (I hoped that this would be different by the end of the lesson), so both went with instinct. Jack could see that he was a big underdog to win, and he’d much prefer to lose $1 rather than $2, so he announced that he would pass the double.

Mary Ann wasn’t interested in the math at all. Her goal was to beat her brother. Since she knew she couldn’t beat him by going with the same answer he gave, she announced she was going to take the double.

Then I went through the math so they would know how to solve these “take-or-pass” backgammon problems in the future.

If they passed the double, like Jack wanted to do, they would lose $1. That much was clear to everybody.

But if they took the double, how do you figure that out?

Well, 10 times out of 36 you win, which would put you ahead $20 on those rolls. Twenty-six times out of 36 you lose, which would put you down $52 on those rolls. Your net loss in 36 rolls is $32, so the average loss is $32/$36 which comes out to 89¢. Since 89¢ is smaller than $1, the correct play is to take the double.

The acronym “QED” comes from the Latin quod erat demonstrandum and means I have shown that which was to be demonstrated — or, basically, this math proves my case. Neither child, however, was impressed by what I had done.

Jack assured me he understood the math, but he would still pass the double. He simply didn’t want to risk losing the extra dollar most of the time.

Mary Ann cared even less for the math. The important thing to her was she got it right and her brother got it wrong! What could possibly be a better result than that? “That was fun! Do you have another puzzle for us, Bob?”

There was nothing more for me to say. In my opinion, playing games successfully depends on understanding and following the math. They both rejected the math. I was out of ammunition.

I spoke to their father, a successful gambler, a few days later about this. I think he took the right approach. He said, “I really don’t care if they become professional gamblers or not. But if they do, I want them to know the math backwards and forwards. They certainly don’t need to know this math when they are pre-teenagers — and who knows what their aptitudes will be in a decade or more? Later, if and when they decide that playing games competently is what they want to do, that’s when it’s important that they learn this stuff.”