[one-more-minute]

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.

[tialaramex]

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.

[function_seven]

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.

[btilly]

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.

[whatshisface]

Maybe once you've started to perceive the meta-patterns between psych experiments, you've taken too many tests to be a good subject.

[tailwind]

"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.

[seoaeu]

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.

[dragonwriter]

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.

[lupire]

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]

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.

[lupire]

If you don't flip the coin within any reasonable definition of flip, sure.

But if you flip a coin and it turns about N times, you can't make the sum (over all k) of the probability of N+2k turns substantially more likely than thr sum of probability of N+2k+1 turns.

[function_seven]

If the mat that my coins are landing on is a strong magnet, I know I can make every single flip land heads. Even when the coin would otherwise land tails, it will instantly flip to align with the strong magnet beneath.

So what if I dial the magnetic field back just a bit? So that only when the coin is oriented flat as it lands will it maintain that orientation in spite of the opposing magnetic forces. But if the coin's orientation is near vertical, then the forces are directed to nudge it "headwards" instead of "tailwards".

Your math applies to weighting the coin. It makes sense in that context. I'm talking about a system of magnetic coin and matched magnetic landing pad.

[shezi]

If you bend a coin, one side has larger area than the other and will prefer to land on that side accordingly. The turn-based argument depends on the fact that both sides of the coin are the same size, which is not true if you bend the coin.

[tedunangst]

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.

[kortilla]

Sometimes, but they have a habit of lying about the purpose.

[tedunangst]

Yes, this is what every very smart person who underperforms or behaves illogically in a study says. Well, actually, I didn't choose wrong, I was testing the experiment. I chose to eat the marshmallow because I wanted to force them to reveal what would happen next, and then they told me the experiment was over, exactly as I predicted. I win again.

[unanswered]

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".)

[frabjoused]

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

[jfengel]

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]

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.

[seoaeu]

> 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."

[tedunangst]

If only there was a link to the study so we could see how it was setup.

[ISL]

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

The PDF is free-to-read.

[blowski]

> 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?

[baobabKoodaa]

> 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.

[vgeek]

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]

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.

[intuitionist]

If they said P(heads) is 60% and you get 4 tails in a row, you also might think the people running the experiment lied to you, especially if it happens near the beginning. But there’s a 13% chance in any sequence of four tosses.

[lupire]

but that means you should bet into the bias, not against it.

[dehrmann]

True; my point was that the person falling for the gambler's fallacy was wrong, but in a sense, so were all the people explaining the gambler's fallacy.

[6gvONxR4sf7o]

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.

[jfk13]

No, the next toss still has a 60% chance of being heads. The coin doesn't remember how it landed last time.

[blowski]

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]

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]

https://dilbert.com/strip/2001-10-25

[blowski]

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?

[ska]

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.

[FredPret]

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.

[baobabKoodaa]

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.

[hansvm]

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]

> 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.

[gen220]

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).

[Jarwain]

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

[curtainsforus]

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?

[WJW]

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.

[EvanAnderson]

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.

[dplavery92]

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.

[quickthrowman]

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]

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.

[Jtsummers]

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]

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]

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.

[dragonwriter]

If you see 60 heads in a row from a coin you’ve been informed is biased to produce heads on average 60% of the time, you'd need a pretty strong bases for trust in your information to not conclude that the most likely explanation is that the bias was underreported. Yes, its possible with the reported bias (or even if the bias was overreported), but that's not the most likely conclusion absent some pretty firm external evidence of the accuracy of the bias estimate you were provided with.

> 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.

People rolling dice aren’t, even when they try to be, perfect randomizers, and with a maximally favorable result and an action which demonstrably repeats it, there’s a strong incentive to repeat the action as accurately as possible rather than even trying to be a perfect randomizer.

[lupire]

I don't believe you.

[Jtsummers]

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.

[koonsolo]

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

[stocknoob]

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?

[deeg]

This wiki page can explain why better than me: https://en.wikipedia.org/wiki/Gambler%27s_fallacy

[JacobLinney]

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]

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.

[sokoloff]

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.

[bscphil]

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%.

[antasvara]

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.

[sorokod]

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.

[mytherin]

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

[justinpowers]

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).

[theschwa]

No, the coin doesn't have a memory, so the chance of tails is still 40% making it still optimal to choose heads.

[afranchuk]

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.

[ska]

> 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.

[skeeter2020]

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.

[prometheus76]

You've just discovered the Gambler's Fallacy.