He’s testing the wrong thing. There have been many experiments in behavioral economics that demonstrate that people don’t invest or bet as a knowledge of probability would dictate. You being unwilling to make this bet more likely means that the benefit you see in winning is less than the penalty you see in losing than it says about what you actually think about probability.
If a coin flipped heads 50 times running, I would doubt the coin was fair or fairly flipped. I once heard a lecture from a magician-turned-mathematician on the difficulty of flipping a coin fairly. I can be done, but requires a lot of practice. He also mentioned that it hardly required any practice at all to flip a coin so that it never turns over and this fact cannot be detected even by a trained magician. I verifed myself how easy it is to do this.
If, however, you replaced coin flips by a random bit generator, then your point is, of course, correct. But I would want favorable odds before I took the bet. Otherwise why should I? The fact that you refused to bet is no evidence at all that your position is wrong.
I wonder if your friend heard of the Monty Hall problem, and is trying to mimic it w/o understanding it.
For the unfamiliar, the monty hall problem does like this: There’s three doors hiding 2 bad prizes and one good one. The contestant “locks in” to one of them by picking it. Monty Hall makes a big dramatic show of opening one of the **other ** doors, which invariably has a bad prize behind it (he’s the host, he knows what is where). Then he offers the contestant to switch his door to the remaining one, or stick w/ the original selection and the show cuts to commercial while the player decides.
Switch or stay? Sound like fair chances? It isn’t. Switch has a 2/3 chance of winning, while stay wins 1/3 of the time. I’ll leave determination of the reasons to you, but consider what I’ve bolded. Didn’t Cecil do an article on this, and get it wrong the first time around?
Anyway, this logic makes it look like past events have bearing on the future, but it’s not true in coin flips. Furthermore, how does the guy know if a coin has been flipped? I mean, what if he tosses it in his pocket? Is that a flip? What about the cash register he got it from? Did that count? What did the previous owner do with it? I find it amusing that someone could flip a coin all day long in an attempt to “burn all the tails up” for later dependence on heads.
Perhaps you did, but I still find it hard to believe. If the coin doesn’t turn, you can see it not turning. Maybe I’m misunderstanding what you’re describing, though?
-FrL-
(Ah: Is it that it flips on more than one axis in a way guaranteeing that the original side always faces up, though the coin appears to be spinning?)
I think the OP’s friend is getting confused. As you flip a coin more times, the ratio of heads to tails will tend to get closer to 1:1, but the actual numbers of each are not predisposed to get closer to one another.
E.g. after 100 flips, you might have 61H, 39T. The difference between the numbers is 22, and the ratio is 61:39
The next 100 could be 52H 48T, you still have more heads than tails, but the total is then 113:87. The difference between the numbers is now bigger (26) but the ratio is more even at 56.5:43.5.
After 10,000 flips you might have 4,902 heads and 5,098 tails. The difference is now 196 but the ratio is closer than 49:51.
This is a version of the Gambler’s Fallacy and your friend is 100% wrong. Even more savvy people can be caught up in subtle errors in this. When I was in graduate school, my stats professor posed the following problem.
“Given that a fair coin toss will most closely predict an equal number of heads and tails over 1,000,000 tosses, if you start the tosses and come up with 10 heads right off the bat, will the results even out over the rest of the trials?”
Most people said yes even though that is incorrect. The most likely result would end up being 999,990 equal results plus 10 additional heads. Trials that have not yet happened are very different than those that have. Probability has no memory and you always start from scratch from where you already are.
If that’s the way he worded it then I’m not surprised people got confused. Yes the results will “even out” in the sense that after 10 flips you have 100% heads and 0% tails, and after 1,000,000 flips (assuming an even split on the remaining 999,990) you will have 50.0005% heads and 49.9995% tails.
The more trials, the closer the results are likely to be to the statistical model. No freaky coin memory required.
Coin flipping is completely memoryless in every way; the ratio is no more compelled to even out in response to past flips than is the difference. It’s true that P(the ratio is within epsilon of 1:1 after N flips) approaches 1 as N increases, but all the same, on each new flip, if the ratio isn’t already at exactly 1:1, then it has a 50% chance of moving towards 1:1 and a 50% chance of moving away from it. (If the ratio is already exactly at 1:1, then, of course, the new flip is guaranteed to move away from it)
I mean, I think maybe you understand all that, but I’m not sure. Bringing up the contrast between head-tail ratios and head-tail differences seems a total red herring.
Hypothetical, you should ask your friend the Monty Hall Problem. Watch his head explode when you tell him the right answer.
As basically mentioned by myself and others above (in posts 4, 18, and 23), I think the fastest way to the end of this, other than via demonstration (which apparently your friend can’t countenance to watch without high stakes riding on it), is to point out to him the ridiculousness of his view, in which undetectably biased coins can be created out of honest coins, without any change to the physical makeup of the coin, just by tossing the coin until a streak of some sort comes up and then pocketing it away for later. I’m sure his explanation in response will be priceless.
You are of course right. And why should you take a zero-expectation bet? Why take risk with no payoff in expectation?
Make him give you odds. Ask him what he thinks the probability of heads is after three tails. 80%? That would mean that 4:1 odds would be a fair bet, so he should be happy to give you 3:1. And if he balks, you should feel free to “sweeten” the deal (by his flawed reasoning) by offering to wait for four heads or tails (or more; who cares, except that you’ll spend more time waiting).
He’s caught in the gambler’s fallacy. Here’s a way to pierce the fog:
Say you flip three heads in a row. The next one should be a tail, right? After all, the odds of four heads in a row is only 6.25%. Bet the farm! Obviously, this isn’t correct, but why? It’s because you’re trying to capitalize on a rare occurence, but the rare occurence already happened. You missed your chance.
After three heads in a row you’ll have a 6.25% chance to flip another heads, 6.25% chance to break the streak with a tail, and an 87.5% chance to have never gotten to this point in the first place. That’s the piece of the puzzle he’s missing.
Once he starts to understand that, see if you can get him to internalize Colophon’s point. Let him think of it this way: You start with 50 heads in a row. It is not reasonable to expect that you’ll ever end up with more tails than heads. After 50 flips, it’s all heads. After 5000 flips, you’ll still have more heads than tails. They don’t keep crossing each other’s path, where some of the time you have more total heads and other times you have more total tails. As the number of flips increase, the percentages get closer and closer to 50-50 but the actual difference keeps growing, making the imbalance larger and larger favoring the same side.
I understand what you said, but it is a fact that the more flips are performed, the closer the ratio (on average) is likely to get to 1:1, precisely because there is no memory of past events.
From the present point, the average results will be 1:1. So if the current tally is X heads and Y tails, the tally in another 100 flips’ time can on average be “expected” to be (X+50) heads and (Y+50) tails, a ratio which is closer to 1:1 than the present ratio.
But this does not of course mean that the coin is in anyway “making up for” the past bias; simply that from the present point you should expect no bias, thus making the previous lopsidedness of results get more and more lost in the random noise.
I hope you get what I mean.
Yes, but if it moves closer, it’ll move the ratio more than if it moves away. So over the long run, there will be a tendancy for the ratio to move towards 1:1.
I think the biggest key in understanding this problem and similar ones is that even though the ratio does tend towards 1:1, it doesn’t get there by cancelling out any past imbalances. It gets there by overwhelming those imbalances. The imbalances will still be there, they’ll just become insignificant compared to the large number of balanced results.
Exactly. Let’s say you start out with 50 heads in a row. Do you know what your expected outcome is if you flip 500 more times? It’s that you’ll finish with 50 more heads than tails, because that’s the point you’re starting from.
That is, if you flipped a coin 500 times, recorded the results, then did it again, and kept doing it, the average result would trend closer and closer to 50 heads more than tails. It’s not that the coin will suddenly begin to correct for past indiscretions, it’s just that from this point forward you expect a 50/50 result, and if you get it the influence of the first skewed result will be diminished.
Oh, I understand that. I just wasn’t sure what the purpose of bringing up the ratio in the first place was, re: the OP’s question, so I had wanted to clear up what I thought was a potential misinterpretation.
You’re absolutely right; this is the biggest key, the point which must be stressed to someone laboring under this delusion, such as the OP’s friend. But the word “overwhelming” could potentially be a bit misleading, the same way “cancelling” would be; the coin of course doesn’t feel any need to apply additional overwhelming balancing power to make up for large past imbalances; just as there’s no additional impulse to come out on the opposite side from the past, similarly there’s no additional impulse to come out balanced in response to an imbalanced past (as you know, but I’m just saying this must be stressed to the OP’s friend). Rather, large past imbalances will end up playing a negligible part in the overall ratio no matter how subsequent flips turn out, whether they turn out balanced or not. That they will probably turn out pretty balanced (in terms of the ratio being arbitrarily close to 1) is its own separate matter, but even without knowing that they will turn out pretty balanced, we can see that the initial however many flips is no stumbling block which must be specifically dealt with, reacted to, and overcome in order to achieve balance in the long run.
(To clarify, I understand Colophon’s purpose in bringing it up now)
You could point out that his reasoning would have been the same after the first 25 tosses - by his thinking there should be more tails from then on to “make up for it”. But as (in your example) you got up to 50 head tosses he would have been wrong.