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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

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.

blowski 5y ago
> 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?
- jfk13 5y ago
  No, the next toss still has a 60% chance of being heads. The coin doesn't remember how it landed last time.
  - blowski 5y ago
    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.
    - ska 5y ago
      Yes, you are wrong, but your confusion is very common. It's so common it even has a name "The Gamblers Fallacy". Over the long run, you expect 40% tails, but if you run the experiment an infinite amount of time there will be sequences of all-heads or all-tails. Because the events are independent, the previous flips don't change anything about what happens next.
      - FredPret 5y ago
        https://dilbert.com/strip/2001-10-25 https://dilbert.com/strip/2001-10-25
      - blowski 5y ago
        So the probablity of 60% assumes infinite flips. Whereas I'm only flipping 10 times, so I won't necesssarily get 60% heads. I'd also need to know the probability that I'm in one of the cases where I get 60% of heads. Is that right?
        FredPret 5y ago
        With a 60%-heads coin you can still get 10 straight heads. It’s just that over many many flips, the average will gradually tend towards 60%. You can still have streaks of hundreds, thousands, millions of either heads or tails in a row.
        ska 5y ago
        This gets into interesting stuff! So the situation described in that paper is that you are given the true odds of the coin, 60% heads. In this case it's just as I described - knowing previous results doesn't tell you anything useful. > Whereas I'm only flipping 10 times, so I won't necesssarily get 60% heads. This is true. In fact there is only about 25% chance of getting exactly 6 of the 10 to be heads (but nearly 70% chance of >= 6 heads). You can work this out with something called the binomial distribution. Chance of getting 10 heads in a row is .6% A more interesting aspect is when you don't know the odds (or don't trust what you've been told). In this case it's definitely important what the history is. So given your 10 flips, we can ask questions like "how likely is it that this coin is fair (50/50) given the 10 flips I just saw". It turns out the best estimation of the true probability is, pretty intuitively, (h+t)/h; this will jump aroudn for small N . In practice you are more often looking at something like P(0.55 < p < 0.65 | samples) , i.e. the probability that the true value lies between 0.55 and 0.65 heads, given the 10 flips I've seen). Obviously in these cases, the more samples you have seen the tighter the estimate get. You can also ask questions like how many flips do I need to see to be confident at a certain the coin is really 0.6 heads.
    - gen220 5y ago
      You’re talking about reversion to the mean, which is a phenomenon that’s related to the law of large numbers. Law of Large Numbers says that, over an arbitrarily large random sampling size, you will eventually end up with a sample that perfectly fits the probability distribution. But the probability of each individual sample is random. This means that, if each sample is randomly-selected and independent, your history of N samples does not affect your N+1th sample. The regression to mean curve is only predictable in the big picture, each bump is 50/50 (or 60/40 in this case).
    - quickthrowman 5y ago
      Previous tosses do not change the outcome of subsequent tosses, so no, it’s not more likely to be tails, it has a 60% chance of being heads.
      - antattack 5y ago
        I understand, once 60% chance is established - then - that's what it is. However, until such probability is established, if I see heads in a row - my intuition would tell me that the physics is skewed towards heads. I don't think that it would be unreasonable to think that in such circumstances until one gets a larger sample of throws.
    - WJW 5y ago
      It's always exactly 60%, no matter how many heads you have already had. That is pretty much by definition, since the problem states that the chances of heads are 60%. In fact, in the real world getting an unlikely string of heads (or tails, or sixes, or whatever) outside of a casino setting probably means that the coin/dice/whatever are unfairly loaded and you should adjust your expectation for the next coin toss even further towards heads.
    - baobabKoodaa 5y ago
      Let's say you're throwing a piece of paper into the trash can from a short distance. Suppose you can successfully throw the paper in the trash 100% of the time. You move your hand. Your hand moves the paper. Gravity pulls the paper down. It collides with the trashcan. It's just a bunch of physical objects exerting forces upon one another. Now, suppose you keep throwing, but somebody has opened a window, so now there's an occasionally gust of wind, which moves the paper in unexpected ways while the paper is in the air. Now you no longer hit 100% of your throws. Sometimes the paper lands in the trashcan, sometimes you miss. Regardless, the paper is still only affected by physical forces: your hand, gravity, wind. Now, suppose you've been really unlucky the past few throws: you have missed 5 throws in a row because of the darn wind. Does it make you more likely to win the next throw, because you are "due" a win? Of course not, because the wind doesn't know or care about your paper throwing hobby. The wind does what it does, regardless of how many of your throws landed in the trashcan. If anything, missing 5 throws in a row makes it _less_ likely to land the next shot, because it may indicate conditions unfavorable to throwing (strong wind, loss of confidence, etc.) Now, the coin flipping experiment with the weighted coin obeys the same physical laws as the paper tossing experiment. It's just a physical object that's affected by forces from your hand, gravity, air, etc. If you throw 6 heads in a row, there's no magic that somehow alters the coin's path in the air on the 7th toss to make it come down tails. The universe doesn't care about our little games.
    - dplavery92 5y ago
      The fact that your previous 6 flips were all heads was an unlikely outcome, but the coin has no recollection of what just happened and doesn't "care" about the past when you flip it again. The maths term for this is to say that each coin toss is "independent." I would not bet that you'd get another 6 heads in a row, but I would bet that the next coin flip will be heads.
    - curtainsforus 5y ago
      You expect 60% of your flips to be heads at the outset. Let's say you flipped it a bunch and you're running at some rate. How could the past flips of the coin possibly influence the flips you get in the future? The coin hasn't changed, the surrounding area hasn't changed, why would the coin suddenly have a different chance of turning up heads on your next flip? There's no probability god that mucks with random chance to make sure 'runs' are balanced overall. Every coin flip is independent, which means all the coin flips are also independent of the past coin flips. If you've "had 60%", that means you've had an unlikely run of heads. Let's say the last 6 flips were 5 heads and a tail, a slightly unlikely outcome (3 in 16, I think). What physical force is acting on the coin to make it less likely to be heads, in the future? Why wouldn't it still have a 60% chance of coming up heads on the next flip?
    - Jarwain 5y ago
      It's not that you're expecting 60% of your flips to be heads, but the coin has a 60% probability of being heads. The former implies that previous flips have an effect on future flips. Or that, if you land on heads 6 times in a row, then the probability of it landing on tails goes up. How would a coin that's weighted to increase the odds of it landing on heads, somehow start landing on tails more frequently? If you flip a normal coin and it lands on heads 10 times, you still have a 50% chance of getting heads the 11th time. The odds of it landing on heads 10 times in a row in the first place is vanishingly small (0.5^10 or 0.097%). But if it Does, the 11th flip still has a 50% chance. The first 10 flips don't affect the 11th. Physically, how Would the first 10 flips affect the 11th? This is all assuming that the coin flips aren't somehow magically linked or casually dependent on each other. The math changes if the previous coin flip could somehow affect the next one. But in a situation where every single roll of the dice is purely independent, then by definition (Because they are Independent ) a previous roll doesn't have an impact on future rolls
    - EvanAnderson 5y ago
      I upvoted you because, while you aren't correct in your assessment, I think this is a good "teachable moment". Human intuition about statistics is really, really bad. I think people who have a better-than-average understanding of statistics forget how bad their intuition is. I suspect it leads to a lot of incorrect assumptions about what a "rational" behavior for someone working from only their statistical intuition would be.
    - hansvm 5y ago
      There are a few other nice answers here, but I think it's important to attack it from as many angles as possible. The intuition that you're going for is that if the true rate is 60% heads and you've seen more than that then to hit 60% odds you _must_ have some extra tails _eventually_. Interestingly, that isn't actually required to make the odds work out to 60% eventually. I'll try for an intuitive explanation: Say you've gotten 10 heads in a row but that the coin really only has a 60% chance of coming up heads. - After 1000 extra flips you'll have 610 heads and 400 tails total on average for a 60.4% chance of heads so far. - After 10k extra flips you'll have 6010 heads and 4000 tails for a 60.04% chance of heads so far. - After 1M extra flips you'll have 600010 heads and 400k tails for a 60.0004% chance of heads so far. Notice how the average percentage of heads is getting closer and closer to 60% even though the extra flips don't have _any_ bias toward tails. A temporary bias toward tails would _also_ suffice, and in much less time (some games like WoW use this for their loot tables I think), but it isn't necessary, and in the example of independent coin flips it does not happen.
    - dragonwriter 5y ago
      > 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? Nope. > 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. Lots of people have explained in terms of independence, which is correct. Another way of looking at it (definitely not more correct, but maybe more compatible with the “a series should eventually match the quoted probability” thinking) is in terms of infinity: If you are expecting 60% of results to be heads, you expect that to hold over an infinite series of flips. If you see any finite number of heads in a row, the probability for each of the remaining flips in the infinite series to get the total to 60% is…still 60%. No finite series of results can change the probabilities necessary to get the infinite series to turn out as expected.
- afranchuk 5y ago
  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.
- theschwa 5y ago
  No, the coin doesn't have a memory, so the chance of tails is still 40% making it still optimal to choose heads.
- skeeter2020 5y ago
  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.
- baobabKoodaa 5y ago
  > 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.
  - 6gvONxR4sf7o 5y ago
    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.
  - vgeek 5y ago
    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.
  - dehrmann 5y ago
    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.
    - lupire 5y ago
      but that means you should bet into the bias, not against it.
      - dehrmann 5y ago
- prometheus76 5y ago
  You've just discovered the Gambler's Fallacy.
- JacobLinney 5y ago
  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.
  - blowski 5y ago
    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.
    - antasvara 5y ago
      The key here is that it's expected to land heads 60% of the time. Take a normal coin, which is expected to land heads 50% of the time. If you flip a heads, do you instantly expect it to be tails next time? By your logic it would be impossible to ever flip heads twice in a row. Coins as a general rule aren't impacted by previous flips.
    - bscphil 5y ago
      That's not a guarantee for any number of flips. For example, if you only flipped the coin one time, what does "60% of the time" even mean in that context? As your other replies have indicated, this is getting at the long-run frequency, meaning as you flip the coin more and more times, approaching infinity, the number of heads approaches 60%.
    - sokoloff 5y ago
      Thought experiment: in what way has it landed heads 60% of the time? It landed heads 100% of the trials so far, but the coin has no way of keeping track of that.
- justinpowers 5y ago
  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) https://en.wikipedia.org/wiki/Gambler's_fallacy).
- Jtsummers 5y ago
  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.
  - travisjungroth 5y ago
    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.
    - Jtsummers 5y ago
      It's certainly low odds, but it's not impossible nor does it require a trick coin. I've seen people roll a 20 on a d20 10 times in a row, and then not a single 20 the rest of the session on the same die. Shit happens, it's probability and it may be improbable but it isn't impossible.
      - lupire 5y ago
        I don't believe you.
        Jtsummers 5y ago
        I mean, that's fine, it's an anecdote. If you'd like, take a few dice and set up cameras and an automatic rolling mechanism and see if there are any improbable sequences like alternation between two or three number or a long run of a single number, or a long run without a particular number appearing. Over enough trials you are likely to encounter these kinds of events.
        deleted 5y ago
        [deleted]
        travisjungroth 5y ago
- mytherin 5y ago
  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 https://en.wikipedia.org/wiki/Gambler%27s_fallacy
- ska 5y ago
  > 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.
- deeg 5y ago
  This wiki page can explain why better than me: https://en.wikipedia.org/wiki/Gambler%27s_fallacy https://en.wikipedia.org/wiki/Gambler%27s_fallacy
- stocknoob 5y ago
  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?
- sorokod 5y ago
  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.
frabjoused 5y ago
I made a little playground for this, you can fiddle with the numbers. https://parsebox.io/dthree/lnumtuenmskr https://parsebox.io/dthree/lnumtuenmskr
tialaramex 5y ago
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.
- lupire 5y ago
  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.
  - function_seven 5y ago
    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.
jfengel 5y ago
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?
- kqr 5y ago
  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+2f%29+%2B+0.4+log+%281+-+f%29%2C+f%3D0..1 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.
- tedunangst 5y ago
  If only there was a link to the study so we could see how it was setup.
- deleted 5y ago
  [deleted]
- seoaeu 5y ago
  > 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."
ISL 5y ago
The paper is pretty awesome and accessibly-written: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2856963 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2856963 The PDF is free-to-read.

blowski 5y ago

> 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?

jfk13 5y ago
No, the next toss still has a 60% chance of being heads. The coin doesn't remember how it landed last time.
- blowski 5y ago
  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.
  - ska 5y ago
    Yes, you are wrong, but your confusion is very common. It's so common it even has a name "The Gamblers Fallacy". Over the long run, you expect 40% tails, but if you run the experiment an infinite amount of time there will be sequences of all-heads or all-tails. Because the events are independent, the previous flips don't change anything about what happens next.
    - FredPret 5y ago
      https://dilbert.com/strip/2001-10-25 https://dilbert.com/strip/2001-10-25
    - blowski 5y ago
      So the probablity of 60% assumes infinite flips. Whereas I'm only flipping 10 times, so I won't necesssarily get 60% heads. I'd also need to know the probability that I'm in one of the cases where I get 60% of heads. Is that right?
      - FredPret 5y ago
        With a 60%-heads coin you can still get 10 straight heads. It’s just that over many many flips, the average will gradually tend towards 60%. You can still have streaks of hundreds, thousands, millions of either heads or tails in a row.
      - ska 5y ago
        This gets into interesting stuff! So the situation described in that paper is that you are given the true odds of the coin, 60% heads. In this case it's just as I described - knowing previous results doesn't tell you anything useful. > Whereas I'm only flipping 10 times, so I won't necesssarily get 60% heads. This is true. In fact there is only about 25% chance of getting exactly 6 of the 10 to be heads (but nearly 70% chance of >= 6 heads). You can work this out with something called the binomial distribution. Chance of getting 10 heads in a row is .6% A more interesting aspect is when you don't know the odds (or don't trust what you've been told). In this case it's definitely important what the history is. So given your 10 flips, we can ask questions like "how likely is it that this coin is fair (50/50) given the 10 flips I just saw". It turns out the best estimation of the true probability is, pretty intuitively, (h+t)/h; this will jump aroudn for small N . In practice you are more often looking at something like P(0.55 < p < 0.65 | samples) , i.e. the probability that the true value lies between 0.55 and 0.65 heads, given the 10 flips I've seen). Obviously in these cases, the more samples you have seen the tighter the estimate get. You can also ask questions like how many flips do I need to see to be confident at a certain the coin is really 0.6 heads.
  - gen220 5y ago
    You’re talking about reversion to the mean, which is a phenomenon that’s related to the law of large numbers. Law of Large Numbers says that, over an arbitrarily large random sampling size, you will eventually end up with a sample that perfectly fits the probability distribution. But the probability of each individual sample is random. This means that, if each sample is randomly-selected and independent, your history of N samples does not affect your N+1th sample. The regression to mean curve is only predictable in the big picture, each bump is 50/50 (or 60/40 in this case).
  - quickthrowman 5y ago
    Previous tosses do not change the outcome of subsequent tosses, so no, it’s not more likely to be tails, it has a 60% chance of being heads.
    - antattack 5y ago
      I understand, once 60% chance is established - then - that's what it is. However, until such probability is established, if I see heads in a row - my intuition would tell me that the physics is skewed towards heads. I don't think that it would be unreasonable to think that in such circumstances until one gets a larger sample of throws.
  - WJW 5y ago
    It's always exactly 60%, no matter how many heads you have already had. That is pretty much by definition, since the problem states that the chances of heads are 60%. In fact, in the real world getting an unlikely string of heads (or tails, or sixes, or whatever) outside of a casino setting probably means that the coin/dice/whatever are unfairly loaded and you should adjust your expectation for the next coin toss even further towards heads.
  - baobabKoodaa 5y ago
    Let's say you're throwing a piece of paper into the trash can from a short distance. Suppose you can successfully throw the paper in the trash 100% of the time. You move your hand. Your hand moves the paper. Gravity pulls the paper down. It collides with the trashcan. It's just a bunch of physical objects exerting forces upon one another. Now, suppose you keep throwing, but somebody has opened a window, so now there's an occasionally gust of wind, which moves the paper in unexpected ways while the paper is in the air. Now you no longer hit 100% of your throws. Sometimes the paper lands in the trashcan, sometimes you miss. Regardless, the paper is still only affected by physical forces: your hand, gravity, wind. Now, suppose you've been really unlucky the past few throws: you have missed 5 throws in a row because of the darn wind. Does it make you more likely to win the next throw, because you are "due" a win? Of course not, because the wind doesn't know or care about your paper throwing hobby. The wind does what it does, regardless of how many of your throws landed in the trashcan. If anything, missing 5 throws in a row makes it _less_ likely to land the next shot, because it may indicate conditions unfavorable to throwing (strong wind, loss of confidence, etc.) Now, the coin flipping experiment with the weighted coin obeys the same physical laws as the paper tossing experiment. It's just a physical object that's affected by forces from your hand, gravity, air, etc. If you throw 6 heads in a row, there's no magic that somehow alters the coin's path in the air on the 7th toss to make it come down tails. The universe doesn't care about our little games.
  - dplavery92 5y ago
    The fact that your previous 6 flips were all heads was an unlikely outcome, but the coin has no recollection of what just happened and doesn't "care" about the past when you flip it again. The maths term for this is to say that each coin toss is "independent." I would not bet that you'd get another 6 heads in a row, but I would bet that the next coin flip will be heads.
  - curtainsforus 5y ago
    You expect 60% of your flips to be heads at the outset. Let's say you flipped it a bunch and you're running at some rate. How could the past flips of the coin possibly influence the flips you get in the future? The coin hasn't changed, the surrounding area hasn't changed, why would the coin suddenly have a different chance of turning up heads on your next flip? There's no probability god that mucks with random chance to make sure 'runs' are balanced overall. Every coin flip is independent, which means all the coin flips are also independent of the past coin flips. If you've "had 60%", that means you've had an unlikely run of heads. Let's say the last 6 flips were 5 heads and a tail, a slightly unlikely outcome (3 in 16, I think). What physical force is acting on the coin to make it less likely to be heads, in the future? Why wouldn't it still have a 60% chance of coming up heads on the next flip?
  - Jarwain 5y ago
    It's not that you're expecting 60% of your flips to be heads, but the coin has a 60% probability of being heads. The former implies that previous flips have an effect on future flips. Or that, if you land on heads 6 times in a row, then the probability of it landing on tails goes up. How would a coin that's weighted to increase the odds of it landing on heads, somehow start landing on tails more frequently? If you flip a normal coin and it lands on heads 10 times, you still have a 50% chance of getting heads the 11th time. The odds of it landing on heads 10 times in a row in the first place is vanishingly small (0.5^10 or 0.097%). But if it Does, the 11th flip still has a 50% chance. The first 10 flips don't affect the 11th. Physically, how Would the first 10 flips affect the 11th? This is all assuming that the coin flips aren't somehow magically linked or casually dependent on each other. The math changes if the previous coin flip could somehow affect the next one. But in a situation where every single roll of the dice is purely independent, then by definition (Because they are Independent ) a previous roll doesn't have an impact on future rolls
  - EvanAnderson 5y ago
    I upvoted you because, while you aren't correct in your assessment, I think this is a good "teachable moment". Human intuition about statistics is really, really bad. I think people who have a better-than-average understanding of statistics forget how bad their intuition is. I suspect it leads to a lot of incorrect assumptions about what a "rational" behavior for someone working from only their statistical intuition would be.
  - hansvm 5y ago
    There are a few other nice answers here, but I think it's important to attack it from as many angles as possible. The intuition that you're going for is that if the true rate is 60% heads and you've seen more than that then to hit 60% odds you _must_ have some extra tails _eventually_. Interestingly, that isn't actually required to make the odds work out to 60% eventually. I'll try for an intuitive explanation: Say you've gotten 10 heads in a row but that the coin really only has a 60% chance of coming up heads. - After 1000 extra flips you'll have 610 heads and 400 tails total on average for a 60.4% chance of heads so far. - After 10k extra flips you'll have 6010 heads and 4000 tails for a 60.04% chance of heads so far. - After 1M extra flips you'll have 600010 heads and 400k tails for a 60.0004% chance of heads so far. Notice how the average percentage of heads is getting closer and closer to 60% even though the extra flips don't have _any_ bias toward tails. A temporary bias toward tails would _also_ suffice, and in much less time (some games like WoW use this for their loot tables I think), but it isn't necessary, and in the example of independent coin flips it does not happen.
  - dragonwriter 5y ago
    > 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? Nope. > 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. Lots of people have explained in terms of independence, which is correct. Another way of looking at it (definitely not more correct, but maybe more compatible with the “a series should eventually match the quoted probability” thinking) is in terms of infinity: If you are expecting 60% of results to be heads, you expect that to hold over an infinite series of flips. If you see any finite number of heads in a row, the probability for each of the remaining flips in the infinite series to get the total to 60% is…still 60%. No finite series of results can change the probabilities necessary to get the infinite series to turn out as expected.
afranchuk 5y ago
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.
theschwa 5y ago
No, the coin doesn't have a memory, so the chance of tails is still 40% making it still optimal to choose heads.
skeeter2020 5y ago
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.
baobabKoodaa 5y ago
> 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.
- 6gvONxR4sf7o 5y ago
  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.
- vgeek 5y ago
  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.
- dehrmann 5y ago
  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.
  - lupire 5y ago
    but that means you should bet into the bias, not against it.
    - dehrmann 5y ago
prometheus76 5y ago
You've just discovered the Gambler's Fallacy.
JacobLinney 5y ago
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.
- blowski 5y ago
  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.
  - antasvara 5y ago
    The key here is that it's expected to land heads 60% of the time. Take a normal coin, which is expected to land heads 50% of the time. If you flip a heads, do you instantly expect it to be tails next time? By your logic it would be impossible to ever flip heads twice in a row. Coins as a general rule aren't impacted by previous flips.
  - bscphil 5y ago
    That's not a guarantee for any number of flips. For example, if you only flipped the coin one time, what does "60% of the time" even mean in that context? As your other replies have indicated, this is getting at the long-run frequency, meaning as you flip the coin more and more times, approaching infinity, the number of heads approaches 60%.
  - sokoloff 5y ago
    Thought experiment: in what way has it landed heads 60% of the time? It landed heads 100% of the trials so far, but the coin has no way of keeping track of that.
justinpowers 5y ago
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) https://en.wikipedia.org/wiki/Gambler's_fallacy).
Jtsummers 5y ago
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.
- travisjungroth 5y ago
  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.
  - Jtsummers 5y ago
    It's certainly low odds, but it's not impossible nor does it require a trick coin. I've seen people roll a 20 on a d20 10 times in a row, and then not a single 20 the rest of the session on the same die. Shit happens, it's probability and it may be improbable but it isn't impossible.
    - lupire 5y ago
      I don't believe you.
      - Jtsummers 5y ago
        I mean, that's fine, it's an anecdote. If you'd like, take a few dice and set up cameras and an automatic rolling mechanism and see if there are any improbable sequences like alternation between two or three number or a long run of a single number, or a long run without a particular number appearing. Over enough trials you are likely to encounter these kinds of events.
        deleted 5y ago
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        travisjungroth 5y ago
mytherin 5y ago
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 https://en.wikipedia.org/wiki/Gambler%27s_fallacy
ska 5y ago
> 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.
deeg 5y ago
This wiki page can explain why better than me: https://en.wikipedia.org/wiki/Gambler%27s_fallacy https://en.wikipedia.org/wiki/Gambler%27s_fallacy
stocknoob 5y ago
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?
sorokod 5y ago
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.

frabjoused 5y ago

I made a little playground for this, you can fiddle with the numbers. https://parsebox.io/dthree/lnumtuenmskr https://parsebox.io/dthree/lnumtuenmskr

tialaramex 5y ago

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.

lupire 5y ago
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.
- function_seven 5y ago
  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.

jfengel 5y ago

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?

kqr 5y ago
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+2f%29+%2B+0.4+log+%281+-+f%29%2C+f%3D0..1 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.
tedunangst 5y ago
If only there was a link to the study so we could see how it was setup.
deleted 5y ago
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seoaeu 5y ago
> 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."

ISL 5y ago

The paper is pretty awesome and accessibly-written: https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2856963 https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2856963 The PDF is free-to-read.