Gamblers fallacy

[js_africanus]

@World Eater

Someone mentioned that he should probably waited for it to get black 3 or for times in a row and then bet on red. Someone said “ah yes, gamblers fallacy”. They said there was a 50/50 chance even after 1000 spins.

Part of the problem as well is that misunderstanding leads to a belief in the Big Accountant in the Sky. As we flip a fair coin, the ratio of heads to tails will get closer and closer to 50/50. Because of this, many people conclude that there is what Garrison Keiler called “odds pressure”: As the discrepancy between heads and tails grows, it becomes more and more likely that the coin will land to so as to shrink that gap. For example, if I’ve flipped two heads and twenty tails, odds are in my favor that I’ll flip a head next time.

Odds pressure is, of course, bullshit. It is the ratio between heads and tails that comes arbitrarily close to 50/50, not the absolute numbers. For example, if I’ve got 7 heads and 3 tails, the ratio is 70/30 and the gap is 4. After 100 tosses, I may have 60 heads and 40 tails, thus the ratio is 60/40 and the gap is 20. After 1,000 tosses, I may have 550 heads and 450 tails, thus the ratio is 55/45 while the gap is 100. You can see that the ratio of heads to tails is getting closer and closer to 50/50, but the discrepancy between heads and tails is getting larger and larger.

Make sense?

Streaks can be disconcerting. So listen to this story. There is a king who wants to win a coin toss very badly. So he decides that he’ll get a ringer. On a certain day, everybody in his kingdom flips a coin. (They each flip one of their own coins, they don’t all flip one giant coin in unison.) About half of those people will flip tails, and are eliminated from the competition. Then the remainder all flip again, about half of whom are eliminated. Then that remainder flip again, and about half are eliminated. This continues untile there are three left, two of whom flip tails and get eliminated. Now the last guy standing has flipped heads, oh let’s say, 50 times in a row. (It’s a big kingdom.) The odds of flipping that many heads in a row are one-in-a-gazillion. The king has his ringer—this new Royal Headflipper can flip heads at will.

Clearly the king is deluded and the Headflipper has about a 50-50 chance of becoming the Headless Headflipper, because he didn’t do anything special. The odds of getting all tails on one of the rounds is pretty low, so it was highly likely that someone would score heads all the way through. This guy just happened to be that person.

There is a book called Conned Again, Watson that you might enjoy. It tackles probability using Sherlock Holmes type stories to go through each quandry.

[js_africanus]

Great stories, Jurph. I wish I typed faster.

[ultrafilter]

This is worth a few more words.

If you flip a fair coin a lot, the number of heads will be very close to the number of tails. People think that there’s some mechanism at work that corrects imbalanced sequences (like 10 tails in a row), but there isn’t: as you flip the coin more and more often, the longest imbalanced sequence becomes insignificant.

In fact, you can show that the probability of getting exactly 50% heads goes to 0 as the number of flips decreases. What is increasing to 1 is the likelihood that you’ll have very nearly 50% heads.

Lastly, the longer a sequence of tails you get, the more likely it is that the coin is biased. 10 is nothing interesting, but 1000 is.

[Mathochist]

@World Eater

Someone explain this to me because I’m missing something.

If I flip a coin and get heads 20 times in a row, why is there only a 50% chance I’ll get tails on the next flip. Isn’t getting heads 20 times in a row fairly rare? How about 1,000 times? Isn’t it more likely that I’ll get tails after 1,000 head flips?

I seem to be missing some simple fundamental point on this.

Because the events are all independant of each other. Yes, the probability of getting 21 heads in a row is tiny (2[sup]-21[/sup]), but the probability of getting 21 heads in a row given that you’ve already gotten twenty heads in a row is just 1/2 again.

[Mathochist]

@Neurotik

Because the odds of you flipping heads 1,000 times in a row are different from the odds that the next one is going to be heads.

There’s always only a fifty-fifty chance of you flipping heads on any individual toss. Each toss is independent from the past toss, so it doesn’t affect its odds.

Quite frankly, though, if you were to flip heads 1,000 times I’d definitely bet heads on the next flip because something is rigging that coin.

Ooh, that gives me an idea for an amazingly fallacious disproof of the Gambler’s Fallacy!

If you get heads twenty times in a row, the law of averages will push the next flip towards tails. On the other hand, Lady Luck is evidently intervening to make the coin flip heads. These two cancel each other out and leave exactly an even chance of either outcome.

[arteitle]

@Telemark

Not directly related to the question but consider the following:

The combination of 10 Heads (HHHHHHHHHH) is just as likely as (HHTTHTTTTH). They are each one distinct pattern out of 2^10 possibilities. Just because all Heads or all Tails looks dramatic, from a probability standpoint it is no more or less likely than any specific random pattern.

This is very similar to trying to explain to someone why playing the lottery with the numbers 1 - 2 - 3 - 4 - 5 - 6 isn’t any worse than playing any other six numbers. Point out that the odds for that specific set of six are exactly the same as the odds for the specific set 1 - 6 - 17 - 28 - 32 - 45, and hopefully they begin to understand.

[MC_Master_of_Ceremonies]

Trying looking at it this way: The probabilty of throwing 20 heads in 20 trials is a bit over a million to 1, the probabilty of throwing 19 heads then 1 tails in 20 trials is exactly the same, infact any permutation is equally probable (or in this case improbable).

Mathocist : perhaps I’m being slightly pedantic and I know that you know this, but the probabilty of throwing 21 heads in a row is dependent on the number of trials considered (e.g. the probabilty of throwing 21 heads in 22 trials is 2^-20).

[MC_Master_of_Ceremonies]

hmm if I’m being pedantic I should make sure my own post is correct to a pedantic level:

“e.g. the probabilty of throwing 21 heads in a row in 22 trials is 2^-20”

[I_Love_Me_Vol.I]

A quick note about the British roulette guy-- he didn’t actually have a 50% (or 1/1) chance of winning when he bet on ‘red’. It was actually an 9/10 chance, slightly worse than 50/50 odds. That’s why the casino can afford to pay out even money on the black and red bets (the times the wheel comes up 0 or 00 essentialy amount to the ‘vig’ for the house).

[Reader99]

i understand the mathematics here. but in the real world, if you observe a roulette wheel and over time it appears to be running to one color over the other, or if a pair of dice seems to be favoring some numbers over others, would it be reasonable to place a bet on that basis? how large a sample would you need to see to account for more than chance results? (to clarify, if it looked like black was coming up a lot on the wheel, i would bet black, not red.)

[ultrafilter]

@Reader99

i understand the mathematics here. but in the real world, if you observe a roulette wheel and over time it appears to be running to one color over the other, or if a pair of dice seems to be favoring some numbers over others, would it be reasonable to place a bet on that basis? how large a sample would you need to see to account for more than chance results? (to clarify, if it looked like black was coming up a lot on the wheel, i would bet black, not red.)

If p is the probability of getting black on a given spin, the probability of getting k black outcomes in n trials is cp[sup]k[/sup](1 - p)[sup]n - k[/sup], for some constant c whose exact form is unimportant.

Using pre-calc level math, you can find that the value of p which maximizes that expression is k/n. So if k/n is significantly different from what you’d expect from a fair roulette wheel (18/38, if I’m not mistaken), you can start to wonder if the wheel is unfair.

[Askance]

@World Eater

Thanks all, I think I’ve got it. A bit of a strange concept to wrap my brain around at first, but the explanations make sense.

What I don’t understand is why this seems a strange concept.

Let me ask the opposite question of you: after 1,000 coin flips coming up heads, what force would act on the coin to make it come up tails? How would it “know” which way up to land, and how would it arrange that it land like that?

[Chronos]

The combination of 10 Heads (HHHHHHHHHH) is just as likely as (HHTTHTTTTH).

To elaborate on why this seems peculiar to many folks: When we look at that first sequence, we mentally describe it as “ten heads in a row”. But when we look at the second sequence, we don’t have a simple, complete description for it, so we mentally describe it as “a bunch of heads and tails jumbled together”. But there is only one sequence matching the description “ten heads in a row”, whereas there are a great many different sequences matching the description “a bunch of heads and tails jumbled together”. So since a jumble is more likely than a pattern, we incorrectly think that any particular jumble must be more likely than a particular pattern.

As for when you should decide that a coin (or roulette wheel, or die) is unfair, that depends on having a prior estimate of how likely it is that the coin is unfair. For instance, suppose I see someone flip a coin heads ten times in a row. The chance of that happening with a fair coin is about one in a thousand. If this is in a very strictly regulated casino, and you inspected the coin ahead of time, you might estimate that there’s a one in a million chance that someone could slip a rigged coin past security. In this case, you would be better off assuming that the coin is fair and the ten flips were just a fluke. On the other hand, suppose that you’re in a third world country where counterfeiting is rampant, and two-headed coins are nearly as common as fair ones. In that case, after ten heads, you could pretty confidently state that the coin is unfair.

Interestingly, Poe in the excerpt quoted by cichlidiot seems to be trying to argue that the Gambler’s Fallacy is true, and that the negation of it is the commonly-held fallacy. This is quite disappointing; Poe was usually smarter than that.

[cichlidiot]

@Chronos

Interestingly, Poe in the excerpt quoted by cichlidiot seems to be trying to argue that the Gambler’s Fallacy is true, and that the negation of it is the commonly-held fallacy. This is quite disappointing; Poe was usually smarter than that.

I agree with you about Poe’s intelligence, but think it made me feel better to know even he had a problem wrapping his brain around the concept, as World Eater stated. At any rate, I do want to be clear, my quote wasn’t an attempt to refute the answers already given. Since I’m posting in this thread again, I’ll also give a nod to ultrafilter, I found his posts in this thread really informative.

[muttrox]

@Telemark

What about the coin landing on edge? Or being grabbed out of the air by a passing parakeet?

What about it? The odds of those are roughly 1 in a kazilliion. The odds of a routlette spin coming up neither red nor black are 2 in 38 (on a wheel with both zero an double-zero. If you think those are equal, I have some interesting wagers to make you…

[Phnord_Prephect]

Please, allow me to throw a nice big wrench into the works of this otherwise logical and correct discussion:

The Not So Random Coin Toss

if a coin is launched exactly the same way, it lands exactly the same way.

If a coin starts out heads, it ends up heads when caught more often than it does tails

Now, with this information, don’t we have to completely rethink the whole concept of coin-tossin’?

I mean, sure, mathematics is nice and all, but it doesn’t adequately explain what happens in real life. GIGO and all that.

If you have a person who is holding a coin that they have just, verifiably, flipped to ‘heads’ 50 times in a row, are the odds still 50/50? This person has found the ‘sweet spot’, or the coin is weighted, or something…

Flip 51: Mathematics says you can’t say for sure. The gamblers say its due for tails. The pattern-finders say it’s gonna be heads.
It’s heads.

Flip 52: Mathematics says you can’t say for sure. The gamblers say its due for tails. The pattern-finders say it’s gonna be heads.
It’s heads.

Flip 53: Mathematics says you can’t say for sure. The gamblers say its due for tails. The pattern-finders say it’s gonna be heads.
It’s heads.

Flip 54: What do you think?

[js_africanus]

@Phnord Prephect

Flip 54: What do you think?

If the coin is verifiably fair? That’s assumption #1. You might also want to assume that the coin is actually being flipped.

I don’t get the impression that the OP is really about precision mechanical accuracy, prodigy-like athleticism, or sleight of hand. The notion that independent probabilistic events are somehow responsible for future outcomes is common enough for me, when I was a copywriter for an online catalog, to have a coworker approach me and critique a text that claimed to give winning strategies on slots. It was the odds-pressure argument dressed up in quasi-mathematical language.

Of coures, given Diaconis remarkable skill for feigning a fair toss, I must say that I’ll be even less likely to bet on a coin flip in the future. Thanks for the link.

[Princhester]

@arteitle

This is very similar to trying to explain to someone why playing the lottery with the numbers 1 - 2 - 3 - 4 - 5 - 6 isn’t any worse than playing any other six numbers. Point out that the odds for that specific set of six are exactly the same as the odds for the specific set 1 - 6 - 17 - 28 - 32 - 45, and hopefully they begin to understand.

This is usually incorrect in that most lotteries, the more people that pick the same numbers as you, the more people you have to share the prize with. So if you pick any recognizable set of numbers (on the basis that, as you say, the odds of that coming up are as good as any other set), you increase the chance that some other smartarse will have done the same and you will halve your payout. The only way to maximise your payout is to pick a set of numbers that no one else is likely to pick.

@AndrewT

What I don’t understand is why this seems a strange concept.

Let me ask the opposite question of you: after 1,000 coin flips coming up heads, what force would act on the coin to make it come up tails? How would it “know” which way up to land, and how would it arrange that it land like that?

The force that lead to such an unlikely event as a coin landing heads 1000 times.

This, to me, is why the gamblers fallacy is so powerful. Others have touched on this before in this thread. Our brains are very very powerful pattern recognition computers. Probably the best ever devised. And most of the time, that is our edge, our intelligence, our evolutionary advantage.

We notice that an animal does something three times in a row. We recognise this isn’t a co-incidence but something we can use, and the fourth time we are waiting with a spear. We recognize that a certain plant grows better every single time if it is in full sun and we then deliberately plant it in the sun and it grows well.

Then a clever casino owner comes along and offers us something that appears to be a pattern (a coin that lands three heads) and our pattern recognition circuit says “I’m on to something here, it will land on heads again” or alternatively “this is too far from the average, it will probably come back to the average soon”.

In both cases, our judgement is correct if there is a pattern behind things, if this pattern happened for a reason. But if it is pure chance, we are calculating on the basis of nothing and we are suckered.

[Telemark]

@muttrox

What about it? The odds of those are roughly 1 in a kazilliion. The odds of a routlette spin coming up neither red nor black are 2 in 38 (on a wheel with both zero an double-zero. If you think those are equal, I have some interesting wagers to make you…

I was just pointing out, hopefully humorously, that even in flipping a coin the possibilities aren’t necessarily 50/50. I’ve heard of folks landing coins on edge. No empirical evidence about the passing parakeet. I know the numbers in roulette are much, much more likely to turn up something that is neither red nor black than a coin not being heads or tails, but the possibilities exist.

[CalMeacham]

The real answer to give to someone perplexed by the Gambler’s Fallacy is this – You may say that the probability, having gotten nine Heads in a row (HHHHHHHH) of getting a tenth Head would be 1 in 2 to the tenth. But what’s the alternative? Ignoring the possibilities of landing on its edge (despite the above post, and an old Batman story I read with Two-Face) or of a passing parrakeet, the only other possibility is nine heads followed by tails, for which thwe probability is – 1 in 2 to the tenth. In other words, the probabilities for the two outcomes are equal. All the “hard work” has already been done by getting the nine heads in a row. And has also been already pointed out, the odds for any other particular combination are just as low. It’s just that most of the other possibilities are unmemorable (aside fropm, a veryu few like nine Tails).