Comment by one-more-minute
4 years ago
Some interesting psychology here:
> In one study, each participant was given $25 and asked to place even-money bets on a coin that would land heads 60% of the time. Participants had 30 minutes to play, so could place about 300 bets, and the prizes were capped at $250.
> Remarkably, 28% of the participants went bust, and the average payout was just $91. Only 21% of the participants reached the maximum. 18 of the 61 participants bet everything on one toss, while two-thirds gambled on tails at some stage in the experiment.
To be fair, as a participant in psychology experiments I go in aware that it's plausible, even likely that I am being misled about what's really going on. That's even necessary in some experiments. Maybe I'm not technically lied to but if deliberately engineering a false impression is the goal, psychologists are the people to do it in a controlled experiment. The experimenters aren't (ethically) allowed to cause you harm, and they'll probably tell you exactly what was really going on afterwards at least if you ask, but during the experiment everything is potentially suspect. Maybe the task you're focused on was just a distraction and they really care whether you notice the clocks in the room are running too fast so that "five minutes" to do the task is really only 250 seconds - but equally maybe the apparent "time pressure" to complete the task is the distraction and they really care whether you lie about completing it properly given an opportunity to cheat.
So if the experimenter in a psych experiment tells me the coin is biased 60% heads, I don't consider that the same way I would if the friend I play board games with says it.
As a result chances are my first few dozen bets are confirming this unusual claim about the world. Biased coins are hard to make, is this coin really biased? Maybe I try fifty bets in rapid succession, $1 on heads each time. Apparently that's expected to take about five minutes of my half an hour, and before that's done I won't feel comfortable even assuming it's really 60% heads.
And at the end of those five minutes on average I turn $25 into $35 and feel comfortable it's really 60% heads or that I can't tell what's wrong.
Now, why gamble on tails? Well like I said, Psychologists mislead you intentionally during experimentation. Maybe the experimenter tells you it's 60% likely to be Heads. If the gamer told me that, I believe it's 40% likely to be Tails because that's logical, but when an experimenter tells me that, I wonder if it's also 60% likely to be Tails if I bet on Tails, and I might be tempted to check.
Spot on.
I kinda feel sorry for psychology and related social science fields. They have an immense hurdle to clear when designing experiments. Both protocol and statistical analysis.
50 or 100 years ago, a study participant might have gone in oblivious to the possibility of subterfuge. Totally unaware that the "taste test" they're participating in for the "marketing majors" was really a study on how political party affiliation affects choices between lemon cake and chocolate chip cookies. Or whatever.
But I have a feeling that college students are much more aware of how these things go today. The experiment is tainted from the get-go by all the participants looking for the "real" data being collected.
I know for damn sure that if I'm recruited for an experiment where I'm taking some sort of test, when a "fellow student" suggests we cheat, that this is an honesty test. Or maybe if the clock runs out before I'm done, I'm being watched for how I handle stress. Wait, is it kind of cold in here? Ah, they must be gauging performance as a function of comfort.
And of course, study participants are way too often 18-24 year olds who happen to go to college. Such a tiny slice of the general population.
So I could see myself placing bets on the "40%" outcome. I wonder if the coordinators straight up told the participants, "Look, we're really testing your betting decisions. This coin really has a 60/40 bias. This isn't a ruse. Please treat this info as true; we're not doing deception testing here" if that would eliminate the kind of second-guessing we're talking about. (I guess we need to study that:) But if that became a norm, then it would further highlight the deceptive tests when that statement is missing.
I feel sorry for social science experimenters.
And of course, study participants are way too often 18-24 year olds who happen to go to college. Such a tiny slice of the general population.
It gets worse. Typically 18-24 year olds who happen to go to the same college as the researcher is working at. So, for example, if this is a large state school then it is a population selected for having SAT scores in a range. Namely above the cutoff to get into the school, but below the cutoff for more desirable schools.
Now suppose that you're doing ability testing. You should expect that any pair of unrelated abilities that help you on SATs will be inversely correlated, because being good at the one thing but landing in that range means you have to be worse at something else. And sometimes that will be the other thing you're looking at.
Several years ago I remember running into a bunch of popular science articles that I found dubious. I tracked down the paper and decided that their analysis suffered from exactly that flaw.
Maybe once you've started to perceive the meta-patterns between psych experiments, you've taken too many tests to be a good subject.
"I wonder if it's also 60% likely to be Tails if I bet on Tails, and I might be tempted to check."
Only if you were clueless, or perhaps if the experimenter said "if you bet on heads it has a 60% chance of winning". Being unstated what would happen if you bet on tails, you might forget that the coin has know knowledge of how you bet, thus making it impossible for there to be any different outcome than a 60% chance of loss by betting on tails.
Even worse, the experimenters didn't actually provide real coins. They just sent around links to a website that they said was simulating a biased coin. Participants presumably had no actual way to know whether the flips were actually 60% biased towards heads, whether the results were truly independent from one flip to the next, or even whether their bet might impact the outcome.
All those sources of uncertainty of the actual probabilities are, while in some cases not typical of a real coin (although uncertainty about actual bias one has been informed of certainly is), fairly typical all of real-world situations in which people face, so I’m not at all certain that that invalidates any application of the results to real-world situations.
Biased coins are *impossible" to make if the coin is flipped not spun.
I doubt any story about a biased coins in the real world.
If the coin was made from a thin magnet, and being flipped onto a weak magnetic plate, couldn't you bias the result? If the landing pad was a strong magnet, then you could trivially make it a "100% heads" coin. Just weaken the magnetic field so it's not strong enough to flip a coin flat at rest, but has enough oomph to take a coin landing near its edge to the preferred result.
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Sometimes an experiment to see if you can go five minutes without eating the marshmallow is just an experiment to see if you can go five minutes without eating the marshmallow, and not a trick to see what happens if they give you three marshmallows after eating the first one.
Sometimes, but they have a habit of lying about the purpose.
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Here's a related yet totally different take: your comment demonstrates flawlessly the reason why sufficiently intelligent people must be weeded out of these experiments (or at least the results). And that in turn helps explain why we end up with people who bet tails.
(Note that the thrill of gambling is another explanation; I'm not claiming "those people are less intelligent, it's the only explanation" but rather "a bias against a certain kind of intelligence could lead to an increase in the observed outcome".)
I made a little playground for this, you can fiddle with the numbers. https://parsebox.io/dthree/lnumtuenmskr
Did they know that it was biased towards heads? With only a 60-40 split I probably wouldn't notice it unless I was actually keeping track, which could take a while. A 6-4 split on 10 tosses doesn't tell you anything. If you told me it was a fair coin and I thought the experiment was about something else, it might take a very long time before it occurred to me to test the hypothesis that the coin wasn't fair.
If they knew it was biased... I'm sure there's an optimal strategy, but a simple strategy would be "bet half of what you have on heads every time". Any idea how much worse that is than the optimal strategy?
You can plot
g = 0.6 log (1 + 2f) + 0.4 log (1 - f)
And locate f=0.5 and compare to the maximum g.
Edit: I wanted to check my intuition so I did: https://www.wolframalpha.com/input/?i=plot++0.6+log+%281+%2B...
Looks like 0.5 is a slight overbet, but still very, very good.
> Did they know that it was biased towards heads?
"Prior to starting the game, participants read a detailed description of the game, which included a clear statement, in bold, indicating that the simulated coin had a 60% chance of coming up heads and a 40% chance of coming up tails."
If only there was a link to the study so we could see how it was setup.
The paper is pretty awesome and accessibly-written: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2856963
The PDF is free-to-read.
> two-thirds gambled on tails at some stage in the experiment
I'm not sure why that's called out. If you've just had 6 heads in a row the next 4 "should" be tails, so it's not irrational to bet on tails is it?
> I'm not sure why that's called out. If you've just had 6 heads in a row the next 4 "should" be tails, so it's not an add thing to bet on tails is it?
I realize you're probably joking, but since this argument is intuitively appealing to many people, I will answer as if it was serious: if you have a weighted coin that is 60% likely to land on heads, that means it's 60% likely to land on heads on any given toss. On the first toss. On the second toss. Any given toss. Even after you have tossed it 6 times and seen 6 heads in a row, the coin is still 60% likely to land on heads. The coin has no "memory". Previous results have no effect on future results.
I quickly searched but couldn't find the exact study, but I've read that by adding the past numbers digital signage to roulette tables, casinos experience a significant (I'm thinking it was like 100%+) increase in wagers when people believe that a color is "due" simply from not understanding independent vs dependent events. Humans love to look for patterns, even when there isn't any real _meaning_ behind them.
There's a corollary to the gambler's fallacy that says is P(heads) is 60% and you get 6 heads in a row, the people running the experiment probably lied to you.
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Moreover, the important feature of coin flips isn’t randomness, it’s independence (from previous coin flips and from everything else). Independence is in fact a useful mental model for randomness.
No, the next toss still has a 60% chance of being heads. The coin doesn't remember how it landed last time.
If I'm expecting 60% of my flips to be heads, and I've already had 60%, isn't it more likely that the next one will be tails?
I'm sure you can probably tell I know next to nothing about either maths or probability, so feel free to explain why I'm wrong.
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That's the gambler's fallacy in action. So long as each event is independent, the prior ones have no impact on the likelihood of future events. If you've flipped the coin 60 times and they've all been heads, there's no reason to expect the next 40 will be tails. They still have better odds of being heads.
If you see 60 heads in a row in the real world you've got a trick coin. The odds of that are 1/10^17.
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I used to drive a fellow RPG-er crazy with this. Whenever I would roll a few times low numbers, I would say "Alright, next time will be high, that is obvious, it's pure statistics!". At first he would object, but still even after he knew that I knew, that statement would still drive him mad.
I remember one time when I rolled really low numbers on a D20, and then there was this really important roll, where I had to get a 20. I confidently said "No problem, I rolled a few really low numbers in a row, so this is definitely going to be a 20, it's pure statistics". Also throwing some calculation in there: "I rolled a 2 and a 1, so in 3 rolls I should get a total of 30 on average, so that means I actually still need 27 to reach the average. That results in more than 100% chance of rolling a 20 right now". And then I actually rolled a 20, was able to keep my cool and a straight face "see, it's just theory". Pure gold! LOL :D
Your friend walks up while you're playing. They haven't seen the game, so think heads is coming up.
Your other friend has been playing longer, before you even started. They saw 13 tails and then your 6 heads. The next throw should be heads to even it out for them.
Why is your history more of an influence than theirs?
This wiki page can explain why better than me: https://en.wikipedia.org/wiki/Gambler%27s_fallacy
Yes it is irrational. That's a common statistical misconception, the key thing here is that every flip has a 60% chance of being heads.
The result of each flip is completely independent of what came before it. In your example the 7th flip is just as likely to be heads as the first flip, or any of the other 5 flips that landed on heads.
It says "a coin that would land heads 60% of the time". If it's already landed heads 60% of the time, I'd expect the remaining 40% for it to land on tails.
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While this is irrational in this experiment, but it is likely that the biological systems in which humans evolved, tend to not have truly independent events - hence our intuition.
The probability of a coin flip being heads or tails is completely independent from the previous flips. If the coin lands 6 heads in a row, the next coin flip still has a 60% chance of being heads, hence it is always unwise to bet on tails in this experiment. This is an example of the Gambler's fallacy [1].
[1] https://en.wikipedia.org/wiki/Gambler%27s_fallacy
Each toss is independent of prior (and subsequent) tosses, so no matter what, a given tosshas 60% chance of landing heads. Rationally, one should bet heads on any given toss.
But most people would agree with the irrational bet. This tendency is known as the Gambler’s fallacy (https://en.wikipedia.org/wiki/Gambler's_fallacy).
No, the coin doesn't have a memory, so the chance of tails is still 40% making it still optimal to choose heads.
Those are independent variables. The fact you've had X heads has no bearing on the future flips. It is irrational to bet on tails statistically speaking, though psychologically that line of reasoning is common.
> If you've just had 6 heads in a row the next 4 "should" be tails
That's not how this works. Each toss is independent, so you should never pay attention to previous results if you know the true odds.
you're not betting on the number of heads/tails per 10 trials though, each trial is independent with a 60% of heads. In a striaght-up prediction you should always choose heads, it the how much to wager that is the question.
You've just discovered the Gambler's Fallacy.