The two men stared at the small cube on the small table.
Ding.
The cube flashed a bright green colour, illuminating the close walls of the dim room, then fading back to the same light-grey dullness as the sky outside the single window.
Thomas pulled himself from the armchair. He looked briefly at John, briefly at the cube, and briefly at the blanketed sky outside.
Ding.
The cube cast a pale red colour across John’s wrinkled white shirt. John’s eyes remained fixed in the direction of the table, as if tied by a thread. Unblinking, thoughtful, – then a jump, eyes to paper, looking down, transcribing a messy, lowercase “r” on a half-filled page.
“Is this part really so important, John?”
John’s neck precessed like a gyroscope, coming to a sudden stop at just the right angle to look directly through Thomas.
“I need to know what the rule is. If I don’t know the rule, I don’t know what it’s for. If I don’t know what it’s for, I don’t which part comes next.”
John’s neck reversed its previous motion. Ding. Red again.
“Edwin is waiting for us. If we don’t get this done in the next two days, we won’t meet him in time to leave.”
John stared at the table. The cube sat on the table. Clouds drifted along outside the windows. Thomas let out a sigh. The cube let out a loud ding and flashed green.
“Are you close to figuring it out?”
John reached a hand into his pocket and pulled out a crumpled dollar. He turned to face Thomas, making eye contact for the first time that morning.
“If the next colour is green, I’ll give you a dollar.”
Thomas frowned.
“Why? You can’t possibly stand to make any money from that.”
“Because I want you to know that I know what I’m doing. So I’m offering odds of one to zero, because I’m that confident.”
Odds can be phrased either as ratios or as fractions.
Thomas frowned. “But you don’t know the rule.”
John stared through Thomas. He looked down at his notebook.
Thomas reached into his pocket and handed John a faded note.
“I’ll buy in. In fact,” – he pulled out another dollar – “if you’re so confident, I’ll pay you if it’s green.”
John blinked. “What? How are you supposed to make money from that? No matter what colour comes up, you’re losing money. That’s odds of one to negative one.”
You should never bet on a zero-probability event if you want to keep your money.
“And I want you to figure it out faster. So I’m paying you. It’s simple economics.”
“It doesn’t work like that. Brains don’t consume dollars and emit formulas. I have to collect data, to formulate falsifiable hypotheses, to-“
Ding. John handed back the dollar and wrote a letter down in his notebook. He closed it in his lap, marking his place with a finger. The thin black lettering on the brown cover announced the obvious – schrijfboek, a writing book.
“Okay,” said John. “You offer me a bet and I’ll decide whether I want to take it.”
“Three to two odds in favour of red.”
If the cube flashes green, he must still pay $3.
If it flashes red, he gets his $3 back, and receives a $2 risk premium.
“You mean, one to two-thirds?”
“No. Three to two. I only have dollar bills.”
“You can’t offer a three-fifths probability of red. They’re symmetric options. This morning, there have been exactly seventy-two of each red and green. Why would you offer anything but one-half?”
The theory of chance consists in reducing all the events of the same kind to a certain number of cases equally possible, that is to say, to such as we may be equally undecided about in regard to their existence, and in determining the number of cases favorable to the event whose probability is sought. The ratio of this number to that of all the cases possible is the measure of this probability, which is thus simply a fraction whose numerator is the number of favorable cases and whose denominator is the number of all the cases possible.
Pierre-Simon Laplace
“You don’t have to take it. I’m just bored. I’ve been sitting here all morning watching colours.”
Ding. The cube flashed red. John glared at the cube, then wrote in his notebook.
“Who knows – if there is a pattern, maybe I’ve picked it up by intuition. In fact – I have a feeling that the next two will both be… green.” He paused. “Or red. Both the same colour.”
“With what odds?”
“Well… I think I’ll go three to six on both red. Although that’s quite a bit. Maybe three to six on both green, just to hedge my bets.”
Placing bets on multiple outcomes increases his chance of making a profit, but not necessarily the amount of profit he should expect to make.
“Why don’t you just put one bet on ‘both colours will be the same’?”
“Okay, fine.” He thought for a moment. “They can either be both red, or both green, or different, so my odds are two to one.”
“One to one-half,” corrected John. “And that’s not quite right. The first could be red, and the second green, or the other way around – you’ve collapsed two different possibilities into one. The odds should be one-to-one.”
He reached into his pocket and pulled out some coins. “Here, I’ll offer you change. You think the probability of both red is three-eighths, and likewise for both green, so your odds are three to five, or three-fifths to one for each. So you give me three-fifths for both red, three-fifths for both green, and I’ll give you a dollar if either happens. I’ll put a dollar on both being the same.”
Thomas stared out the window. He looked at the silent grey cube. He glanced back at the armchair. His eyes crept back to John.
“What?” Thomas sighed again. “Okay.”
A loud sound erupted from the armchair. Thomas stepped back in shock, then realised that John had begun laughing.
“That’s a terrible deal,” began John. “No matter what happens, you’re always going to lose money. Look – you pay me twice three-fifths at the start, and I give you a dollar, so I’m already up one-fifth to begin with. Then, if they’re different, no money changes hands, and if they’re both the same, we’ll each pay each other a dollar. If you go around accepting bets like that, people will take all your money!”
Thomas thought for a second, then shook his head. “The point is to make you think about the cube. This isn’t thinking about the cube. This is thinking about the money I’m giving you to think about the cube.”
John didn’t seem to notice, and continued explaining.
“No matter what odds you offer me, unless the probabilities of events like both colours are red and both colours are green add up to the probability of both colours are the same, I’ll always be able to guarantee you lose money. If you bet, that is.”
“What if I don’t?”
Events that cannot occur together are called mutually exclusive, and the probability of both is zero.
Ding. The cube flashed green. Ding. The cube flashed red. Both men stared, startled by the quick succession. John wrote down two letters into his notebook. He frowned and flipped back a page. Then forward a page. Then back again. He counted.
“The number of reds and greens is about the same, but…”
“But?”
“Well, that makes three times today where the cube has flashed red, green, red in succession. Yesterday, it did that five times. Of those eight times, seven of them have been followed by red. Six of them have been preceded by green.”
Ding. The cube flashed red.
“Does that matter?”
“Well, normally, for the next three flashes to be exactly red, green, red, three flashes need to all be correct. Each has a one-half chance of being correct. So if we suppose they’re all independent, there should be a one-eighth chance of those being our next three colours.”
Thomas was staring out the window again, but he had been listening. He hated that smug look John got when explaining something. He hated how he always overexplained the obvious – or underexplained the need for complexity. But John’s explanations had an occasional, infuriating tendency to be simultaneously sensible and insightful.
“Suppose we ask, ‘will the next three colours be red, green, red?’ Then we have to answer ‘yes, with probability one-eighth’. This matches the records I have. Now suppose I ask you, ‘will the next four colours be green, red, green, red?’ The last three colours on their own seem to have a one-eighth chance, but once we know that, the first green jumps from a one-half chance to a three-quarters chance.”
“But that’s using the future to predict the past. Anybody can do that. How is that helpful?”
“Well, now we know that for a sequence of four, the probability of the first colour being green given that the last three are red, green, red is three-quarters.”
“And?”
“So what’s the probability of some four colours being precisely green, red, green, red?”
“Well… The first colour is still just one-to-one odds. But I guess if we were watching the cube flash in reverse, we’d think that it would have three-to-one odds in favour of green.”
“Yes.” said John. He looked a little disappointed at his failure to adequately confuse Thomas. “But the probability of the reversed sequence of four coming from the reversed cube should be the same as the probability of the original sequence of four coming from the regular cube.”
Thomas thought for a moment. “So I guess… well, working in reverse, we should see the red, green, red triplet one-eighth of the time. And then, once we have that, we should see the green three-quarters of the time. There’s nothing more we can say about it that than so far.”
John looked down at his book, then back up at Thomas. Now he was confused. Since when had Thomas known probability theory?
John shifted in his seat. “Well, from what you’re saying, that group of four should occur three-sixteenths of the time, rather than just one-sixteenth.”
“Yes.”
“We know the probability of the first green and the last three is the probability we’d give to the next colour being green, times the probability we’d give to the next three being red, green, red after seeing that first green. We also know that’s the same as the probability of seeing red, green, red in reverse, times the probability we’d give to the first of the four being green after learning those last three.”
Thomas was confused again. “I’m confused again,” said Thomas.
“Here,” said John, pleased. “I’ll write it down.” He turned to the last page of his notebook and wrote down an equation. “B is the first colour being green, and A is the last three being red, green, red. The vertical bar before B means ‘given that B is true’.”
Thomas lifted the book and stared at the formula. “So if we’ve just seen a green, the probability of red, green, red coming next should be… the probability of those three occurring before we saw the green, times the probability of seeing the green before those three, divided by the probability of seeing the green?”
“Uhh,” said John. “Probably.”
Ding. The cube flashed a brilliant shade of blue. Both men stared.