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Something fun on a lazy day. I was asked to comment on this tweet:

Imagine a circle laid flat on a table, with a little spinner on it that you can flick. The circle is 80% red and 20% yellow, like a yellow pie slice.

You flick the spinner. It spins, and slowwwly stops.

Where did it land?— Aella (@Aella_Girl) November 20, 2022

Before we get to it, and if you can’t see the first, because I believe the poll is still going as of this writing, there is also this older one:

Imagine a circle, with a little spinner on it that you can flick. The circle is 80% green and 20% purple, like a purple pie slice. You flick the spinner. It spins, and slowwwly stops. Where did it land?

— Aella (@Aella_Girl) January 3, 2020

And just in case Twitter itself craps out (a dice metaphor), here’s the words:

Imagine a circle laid flat on a table, with a little spinner on it that you can flick. The circle is 80% red and 20% yellow, like a yellow pie slice.

You flick the spinner. It spins, and slowwwly stops.

Where did it land?

About 80% voted red, the remaining voting yellow.

One commenter said, “The poll results pretty closely matches the 80/20 probability of each result. Super cool kinda a wisdom of crowds moment.”

Is this close match the wisdom of the crowd, or something else?

All probability is conditional. Given the premises about the spinner, the probability the spinner stops on red is indeed 80%, as we can all see, making the probability of yellow 20%.

That probability does not say which color the spinners stops at. It is only a probability, and nothing more.

The probability can, and does, function as a prediction of the spinner. It says not which color is final, but gives the chances. Predictions—also called forecasts or scenarios—are best put in terms of the full uncertainty warranted by the information specified.

To make a bet (which is what at least most voters did) is to take the prediction and turn it into some kind of “action.” Which can be entirely counterfactual, as it is here. There is no circle, the spinner won’t be flicked, we can only imagine it.

Now some people might not have been making a bet, and might have seen the close match and decided, for fun, to juice it along by voting for the color that currently had a deficit, or even damage it by voting for the color that currently had an excess—and where they can glean the excesses and deficits before they vote by the comments to the tweet. We can’t know. Just as we can’t know a host of other possibilities.

How did people make their vote/bet? One man said, “I imagined it actually was a slice of lemon pie in a pan with a spinner. Since it stops slowwwly, I assumed the spinner got stuck in the pie. So yellow. :)”

Others reported similar experiences. The end was the close match between the frequency of bets and the deduced probabilities.

Which is how we might expect it should be—*if* people are good at deducing probabilities in situations like this, which they obviously are, *and*—this *and* is the crucial point—in which most or all have the same stake in the outcome.

Everybody “paid” the same to vote, everybody who “wins” would win the same amount, and the same with loses. Whatever would be won or lost is also the same for the future prospects of all involved. What I mean by that is this: a billionaire and a hundredaire will win or lose more proportionally than each other on a money bet, which affects their future selves differently, even when both guess the same and have the same outcome.

Not only that, but everybody also knows all these things, or at they least weakly intuit them, just as they know they can safely bet, because there will be no real winners or losers, and thus no reason to crow or to receive trash talk and the like.

We have a nice and sweet uniformity of the kind found in textbook homework cases, but rarely or never in real life. And we have a simple, even trivial, deduction of the probabilities.

From all this we have to guess *why* people bet/voted the way they did, and in that proportion.

We have two experiments and their details (the older tweet), and the comments of some of the participants.

Now I voted red because red is the most likely outcome. Many surely did the same. But why didn’t everybody? Why, in other words, did 20% choose the less likely outcome when that 80% is the probability-optimal bet?

A similar, but not equivalent, question is why do people at casinos bet less likely outcomes when more likely options are available (here’s a nice example showing many bets people shouldn’t be making, but do). It’s not exactly the same situation because of the influence of bet size, as discussed above.

But it’s not entirely different either. People on average bet slots far more than they bet blackjack, when blackjack has far better odds. Some of this has to do with desiring to be left alone versus those who like to bet with people. Some like the press the buttons. And some has to do with the smaller usual bets on slots than on blackjack.

Even still, even with bets that are more equivalent in terms of style of play—cards among people, say, versus single electronic machines—not all flock to the best bets. But since money is always involved, we have to be cautious with the evidence.

It’s difficult to find evidence, too. People have looked into these things, but finding stats on frequencies of bets by their known probabilities/odds is hard to impossible. Casinos don’t appear to keep track of the precise number of bets made in non-electronic games (which we’d need).

Here’s a picture of what I think is happening with the human psychology:

This is only a cartoon, the S-shape of the black line might be more exaggerated in real life. Here’s how to read it.

The x-axis shows the known probability of “minority” bets, i.e. those less than 50%. The y-axis shows the frequency at which non-consequential bets would be made. The dots represent a one-to-one line, which are when the minority bets match the bet frequency.

I think the 20% minority by 20% frequency is a “sweet spot”, something like the old “80-20” rule. My guess is that when the minority is 50% people would also bet with a frequency of 50%.

But I think the frequency below the 20% minority does not fall linearly, and instead falls much slower, and would level out around some “high” number, say, 1 to 5%.

In other words, when the spinner has only 1/1000 yellow or 1/10,000 yelow, about 1-5% of people would pick it, and not 0.1% or 0.01% of people. People bet lotteries like Powerball, which has minuscule probabilities of winning, but where the proportion of people who bet is much, much larger, even when the jackpot is relatively small. (Though even there we can’t discount the effect of money on whether to bet.)

For probability values larger than about 20%, I think more people would bet closer to 50%, becoming exactly 50% frequency at 50% probability. (Unless hateful colors, or whatever, are chosen for the minority bet.)

In other words, I think the near—but not exact!—match at 20% was a bit of luck. Something like the Pareto principle of non-consequential bets.

My guess (my model!) is based on behavior at casinos and lotteries. It’s easy enough to check. But we have to check on people who don’t know of the original tweet, and don’t know of this article.

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Categories: Fun, Statistics

Interesting article! I wonder what the results would be if you modified the question slightly…

“Imagine a circle laid flat on a table, with a little spinner on it that you can flick. The circle is 80% red and 20% yellow, like a yellow pie slice. If the spinner lands on yellow, you win (only in your imagination!) $20. If it stops on red you win nothing, but it costs you nothing. You flick the spinner. It spins, and slowwwly stops.

Where did it land?”

Would more people vote for yellow, just because it now has an imaginary payoff?

Nolan,

Almost certainly.

The pattern you identify appears to be the well-known favourite/longshot bias. https://scholar.google.com/scholar?hl=en&as_sdt=0%2C5&q=favourite+longshot+bias&oq=favourite+long-shot+

> The end was the close match between the frequency of bets and the deduced probabilities.

Must disagree with that. With a sample size of 3930, the empirical result is 264.4 billion* times more likely to come from the empirical mean (0.765) than from the theoretical mean (0.80), which even people who avoid loaded words like “significance” might consider strong evidence against.

(Of course the right choice would be to pick red, but we know from BDT that human people match odds, odd as that behavior may be. 😉

*Pr(data|p =0.765)/P(data|0.8) = 2.64433E+11, using back-of-the-envelope calcs. 😉

Silva,

All her other attempts had rates higher than 80% for majority. I only said “close match”.

And probability doesn’t exist. And the theoretical mean for the bet is 100% majority, not 80%.

Briggs,

Probability exists as much as a Banach space, symplectomorphism, Aleph-2, the set of all sets that don’t contain themselves, and every other mathematical construct past arithmetic: it’s a tool for thinking. A power tool that can harm those who don’t know how to operate it or are careless in its use, but a tool nevertheless.

The theoretical mean is for the realization of the spin, not the optimal decision given the setup of the problem, as indicated by the quoted part “frequency of bets and the deduced probabilities.”

J

Silva,

Sure, in those senses. But not as it’s used to indicate a true value of some parameter at infinity that we estimate finitely.

No, the theoretical mean is the optimal bet. We wondered why the difference, a matter of psychology. Your math indicates results in the direction opposite of what we see, which is greater than 80% for the majority bet.

Briggs,

My LR test is simply whether the 76.5% indicates odds-matching to the 80% spin or not. It tests a known result from BDT that people odds-match, even when taking the dominant action would be the right answer. Given that it’s 1/4 trillion against, I’ll go with “no.”

As for probability and its value, there’s a Portuguese saying that is hard to translate but is something like “in life, even the smallest advantage is better than no advantage at all.”

Interesting phenomenon. I have several guesses as to why empirical distributions turned out to be consistent with the expected one in this case. Here is one of them. Approximately 60% of respondents clicked red based on probability. (Perhaps, 60% is optimistic on my part. ) The other 40% chose randomly, and half of them chose yellow.