Thursdays at the Whetstone: The Two-Dice Table
On Thursdays the back table at the Whetstone belongs to the regulars. Nobody booked it. It is simply understood. There is a pot of tea, a plate of the almond biscuits nobody admits to ordering, and, this Thursday, two dice and two cups.
Nana Fives had brought the dice. She had been a dominoes player for most of her life, the kind who counts the open ends in fives without moving her lips, and she had lately taken up the dice "to keep my arithmetic honest."
"Here is a fact," said Nana Fives, rattling one cup. "Throw two dice. Seven comes up more than any other total."
"Everyone knows that," said the Captain. He was out of uniform, in a cardigan with a button missing, and he looked smaller without his brass. "Seven is the lucky number."
"It isn't luck," said Nana Fives. "It's counting. There are thirty-six ways two dice can land, if you keep them apart in your head, a red one and a white one. Six of those make seven: one and six, two and five, three and four, and the same three turned round. No other total has as many. Two has one way. Twelve has one way."
"Then I'll back seven," said the Captain, and pushed a biscuit into the middle of the table. "Seven on the next roll. A biscuit says so."
Nana Fives rolled. The dice stopped on a five and a three.
"Eight," said Inkling's grandmother, who had not yet said anything.
"You said seven was the most likely," said the Captain, aggrieved, and took his biscuit back.
"It is," said Nana Fives. "Did I say it was likely?"
The Captain opened his mouth and closed it again. Then he did what he always did when he was losing an argument, which was to state the other side better than its owner had.
"All right," he said. "Let me say what you mean, and you tell me if I've got it. Of the eleven totals two dice can make, seven is the most common. But six ways out of thirty-six is one in six. So most of the time, five rolls in six, the dice come up something else. Seven is the most likely total, and it is still unlikely on any one roll."
"That's it," said Nana Fives, pleased. "Better than I said it."
"I know," said the Captain. "It's my only talent."
They rolled a few more. A nine. A six. A seven, which the Captain greeted like an old friend. A four. An eight again.
Inkling's grandmother had been listening the whole time, the way she did, with her hands round her cup. She listened longest of anyone at the table. When she spoke, people tended to stop rolling.
"Here's what I want to know," she said. "My granddaughter plays a game with dice on Saturdays. When seven hasn't come up for a while, she says it's due. She gets quite cross with the dice. How long is a while?"
"How do you mean?" said the Captain.
"I mean, if seven comes up one roll in six, how often do you go six rolls with no seven at all? Is that rare? Should she be cross?"
Nana Fives worked it out on a napkin. The chance a single roll isn't a seven is five in six. For six rolls in a row, you multiply that by itself six times.
"About a third," she said, and turned the napkin round so they could see. "Five-sixths, six times over, is a little more than one in three. So one evening in three, if you roll six times, you'll see no seven at all."
"That often," said Inkling's grandmother.
"That often."
"So a long wait for seven," said the Captain slowly, "isn't a sign the dice are broken. It's just an ordinary evening."
"And the dice don't know they owe you one," said Nana Fives. "That's the part people find hard. Each roll starts fresh. The dice have no memory. A seven is no more due after six misses than it was before the first."
The Captain leaned back. "Then here's my difficulty," he said. "If I'd waited six rolls with no seven, I would feel, very strongly, that one was coming. I wouldn't be able to help it."
"Of course you would," said Inkling's grandmother. "Everybody would. That's why it's worth saying out loud." She picked up the dice and turned them over in her palm. "I don't think the lesson is that the feeling is wrong. I think the lesson is that the feeling isn't evidence."
There was a pause while everybody decided whether they agreed with that.
"It's the same at the dominoes table," said Nana Fives after a while. "When I was younger I'd sit with a hand full of fives and wait for the double five to come out, because it felt like it must. I'd hold my best tile back for it. Sometimes the whole game went by and it never came. It was in the boneyard the entire time, face down, not knowing it was wanted."
"Did you stop waiting for it?" asked the Captain.
"I stopped holding tiles back for it," said Nana Fives. "That's not the same. I still wanted it. I just played the hand I had instead of the hand I was owed." She tapped the napkin. "That's what this sum is for. Not to stop you wanting the seven. To stop you spending your biscuits on it."
"A useful sum," said the Captain, and ate the biscuit he had nearly lost, to be safe.
"So which of us was right?" asked the Captain. "About seven."
"You were both right and both wrong," said Inkling's grandmother. "Nana was right that seven is the most likely. You were wrong to bet on it once. And my granddaughter is wrong that it's due, though I shan't tell her so in those words."
"What will you tell her?" asked Nana Fives.
"I'll give her the dice and a pencil," said Inkling's grandmother, "and I'll ask her to roll thirty-six times and keep a tally. She'll find seven at the top. Probably about six of them, give or take. And somewhere in the middle she'll find a run of six with no seven in it, and she'll be cross, and then she'll look at the tally and see that seven still won."
"And if she doesn't find a run like that?"
"Then she'll find one next Saturday," said Inkling's grandmother. "One time in three. Isn't that what you said?"
The Captain laughed and pushed the plate of biscuits into the middle of the table. "Then I withdraw my bet," he said. "I'll bet on the tally instead."
The question for your own table
Was the Captain wrong to back seven? Not wrong about which total is most likely. Wrong about what "most likely" lets you expect from a single roll. Before you read on, decide what you would have bet, and why.
Something to do
Two dice, a pencil, thirty-six rolls. Keep a tally of every total. Look for the longest run without a seven. Then compare the tally with the count of ways: one way for two and twelve, two for three and eleven, up to six for seven.
Across the table
The dialogue in three lines. Nana Fives says seven is the most common total of two dice. The Captain bets on the next roll, loses, and then states her point better than she did: most likely is still unlikely, about one roll in six. Inkling's grandmother asks how often six rolls go by with no seven at all, and the answer, about one evening in three, changes the conversation.
One question to ask. "If something is the most likely thing, does that make it likely?" Try it with weather, buses, or which friend calls first.
One thing to do at the table. Roll two dice thirty-six times and tally every total. Mark the longest stretch with no seven. Talk about how it felt to wait, and whether the waiting changed the next roll.
The grown-up layer. The children's version of this idea is in Gullwick's first book, where Lucky the ladybird tells Lan to search the seven patch because seven has the most ways. The grown-up version adds the harder half: a long wait is ordinary, and the feeling that a seven is "due" is the gambler's fallacy.
The Honest Page
- Known. Seven is the most common total of two fair dice: six of the thirty-six equally likely pairs.
- Known. So seven still comes up only about one roll in six, and five rolls in six are some other total.
- Known. Six rolls in a row with no seven happen about one time in three.
- Disputed. Whether people feel a seven is 'due' after a long wait because of how memory works, or because of how the question is asked, is argued about by psychologists. (source)
- Legend. That dice remember what they rolled last. They do not, but the feeling that they do is very old.
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