Gambler's fallacy vs Regression to the mean
How are both different?
If we get ten heads in a row when we toss a coin, we may think a tail is due, which is the gambler's fallacy. Since the coin has no memory there's an equal probability on the eleventh toss for either heads or tails.
But regression to the mean says things will average out in the long run. Even this assumes that the coin has no memory, no? If it is a fallacy to think that a eleventh toss is due, is it correct to think knowing what happened in a thousand or ten thousand tosses and betting that a tail is due.
Does it depend on the size of the sample space? If we are in the tail end of a big sample space then would the gambler's fallacy hold true? Or does some other property make them different?
September 26th, 2006 3:58am
'If we are in the tail end of a big sample space then would the gambler's fallacy hold true?'
Yes it will. Because of it being a large enough sample, the avarage will already be at 50%.
The next toss being heads or tails will not make a difference to the avarage, even if the preceeding tosses have all been tails for instance.
September 26th, 2006 4:34am
... Since the coin has no memory there's an equal probability on the eleventh **zillionth** toss for either heads or tails. ...
September 26th, 2006 4:37am
Regression to the mean says that over ten thousand throws, you're likely to get a 50/50 distribution. If you have ten thousand heads in a row, then the odds are that, on the macro scale, the next then thousand will be mostly tails.
But it doesn't say anything about what the *next throw* will be.
I agree with the coin not having memory if it was all tails or even if it is the zillionth time. But if we agree with regression to the mean should there be a change sometime? I want to know if the gambler's fallacy is only part of the equation.
Like if the preceding tosses have all been tails, then, doesn't the law of averages (or regression to the mean) say that they have to go towards being more equal?
"For example, after tossing a "fair coin" 1000 times, we would expect the result to be approximately 500 heads results, because this would reflect the underlying 0.5 chance of a heads result for any given flip."
Steele - So wouldn't the next x number of throws at least have more chance of being tails. Then the gambler's fallacy may not be very wrong. In the sense, knowing what we know what happened before it would be better bet on tails. It then wouldn't be as incorrect as we do it when there were only 5 heads in a row and we bet the sixth to be tails. And in both the cases the coin has no memory. But in the first case, law of averages would say that it is not exactly an equal probability.
If the law of averages is true, though the coin has no memory, wouldn't the chances of the next toss being a tail be more?
September 26th, 2006 5:04am
If you have 10,000 heads in a row, it is virtually certain that the 10,001 toss will be heads also.
September 26th, 2006 5:11am
John - Wouldn't that also be similar to the gambler's fallacy where we think that the coin has memory but will do the reverse? If the coin has no memory, then another tails is not 'due'. In the same way, another heads may not 'continue'. That is, if someone thinks a tails is due will be virtually certain because so many heads went by. If they think that heads will continue, like you do, since so many heads went by, they will be vitually certain that the next is also likely to be heads.
If we go by the gambler's fallacy, the odds are even. But if we look at the law of averages, the odds seem to be more towards tails. But both assume that there's no memory.
September 26th, 2006 5:52am
As an aside: How would one go about determining if a coin is "fair" if it came up heads 10000 times in a row?
Anyway, to the original question, you need to take into account when you made the "bet". If you bet that the next ten tosses come up heads, the odds are 1/2^10, if you're betting on the next toss the odds are 1/2 regardless.
>If you have 10,000 heads in a row, it is virtually certain
>that the 10,001 toss will be heads also.
No, it's virtually impossible for heads to come up 10,001 times in a row, but the odds of the 10,001st toss are 50-50.
nope. if you have flipped 10000 heads in a row then the coin is weighted and the next toss _will_ be heads.
September 26th, 2006 6:19am
and if you just threw 10000 tails in a row, it seems unlikely to be a fair coin ....
September 26th, 2006 6:19am
ah. wehat wsv just said. sorry.
September 26th, 2006 6:19am
tim - I agree with the probabilities you have assigned for the ten tosses and the very next toss. But my doubt was whether the knowledge of the law of averages will help us alter the odds so that a favorable bet can be made.
We'll assume that the coin is not biased. It just gives more heads than tails for a lot of tosses instead of a handful few. The probability of the next one toss and the next ten is what tim has given above.
If the next toss we were going to make was the very first one or the ten the first ten it's fine, everything works as we have discussed. There's no long run that has happened here to warrant the use of the law of averages. But if there's been a long run and as a fair coin, the numbers of heads, by chance, have been more how would the law of averages affect the outcome.
If we assume 1000 tosses to be long run for the law of averages to apply, the number of heads and tails will tend to be around 500 each. They will not be exactly that, obviously, but they'll tend to it. Suppose also that there's been 600 tosses and the number of heads is 400 and tails 200. The probability of tails on the 601s toss is 1/2 without the law of averages taken into account. If it is, wouldn't it be different?
A person under Gambler's fallacy may win here, though, he would be right for the wrong reasons. That is, he thinks that a tails is 'due'. But the coin has no memory but the law of averages takes over. If he had thought it to be biased and that a heads will 'continue' he would have lost.
September 26th, 2006 9:18am
People have said this, but I'll add:
"Regression to the mean" is a way of measuring the "fairness" of the coin. The "Gamblers Fallacy" ASSUMES a 'fair coin' to begin with.
If *I* had a coin, that gave me 10,000 heads in a row, I would SERIOUSLY doubt the "fairness" of that coin. It *IS POSSIBLE* for a 'fair coin' to do that -- but that possibility is very tiny. Shoot, after 100 heads in a row, I'd be asking questions about that coin.
So, every 'toss' instance is independent of every other instance. If it's a 'fair coin', it has a 50/50 chance of heads or tails on each 'toss'.
September 26th, 2006 9:18am
This seems to cover both fair and biased coin flipping
Interesting - Schilling did most of the original theory cited but the articles in an archive. :-(
September 26th, 2006 10:05am
> Regression to the mean
People do become nastier over time.
son of parnas
September 26th, 2006 10:28am
Gambler's fellacity ... over time, gamblers resort to giving their brokers blowjobs.
yes we do
September 26th, 2006 3:19pm
----"The probability of tails on the 601s toss is 1/2 without the law of averages taken into account. If it is, wouldn't it be different?"----
No; the law of averages merely says that the number of heads and taila will be approximately equal on average.
That is to say your set of 20,000 throws doesn't have to follow the law of averages, it merely contributes to it.
September 26th, 2006 3:39pm
'As an aside: How would one go about determining if a coin is "fair" if it came up heads 10000 times in a row? '
Apply Bayes Rule.
September 26th, 2006 3:49pm
trollop - Thanks for the link. Somehow I didn't think to look into Runs. When I searched that way I got the following link which gave a similar example to what I was thinking.
In the section 'The law of large numbers / "The law of averages" ', the first line tells that the theory of probability becomes of enhanced value to gamblers when it is used with the law of large numbers. I guess this was what I was driving at.
It gives an example of 100 tosses first and 100 more tosses and how, though law of averages works, which he calls a 'corrective action' setting in, one may make a bet which can turn out to be wrong.
Stephen - I agree that throws don't follow the law of averages but that it merely contributes to it. Though, it may sound like I mean that it follows the law of averages, I meant that it contributes to it. It has no memory. But knowing that it contributes to it, I thought we should be able to alter the odds with the knowledge.
If we think it follows the law, we may place our bet on the next toss without knowing the distribution of the previous tosses. But if we know that it contributes to the law, we acknowledge that the law of averages will work out even if the way the tosses go to obey the law may be as slow as logarithmic growth.
The above link says "As the number of tosses gets larger, the probability is that the percentage of heads or tails thrown gets nearer to 50%, but that the difference between the actual number of heads or tails thrown and the number representing 50% gets larger".
September 27th, 2006 2:13am
Slightly relevant :-)
Note in most casinos a run of 5 odd pairs (a head and a tail) forfeits all current bets to the house - this house "edge" of a little over 3% sends smart gamblers off looking for a roulette or baccarat table.
September 27th, 2006 9:54am