Is the Gambler's Fallacy Really a Fallacy?

[Volkl]

If I was a serious gambler there is a small chance that I could place 50,000,000 bets physically at a casino. Pretending that the game we are playing offers fair odds, the chances of one particular outcome coming up 1000 times in a row within the set compared to the same particular outcome coming up 100 times must be much less. If this is true the probability of 20 particular outcomes in a row must be less then the probability that 10 of the same particular outcomes can come up. Doesn't this prove that there is a tendency towards randomness meaning that there is a tendency to have less of the same particular value coming up in a row. To me, this logic proves that the gamblers fallacy is in itself a fallacy. Or do you believe that all 50,000,000 could be the same value for anyone living on earth? I had a roulette wheel with no greens in mind.

 

[micromass]

No, let B=black and R=red. Then the probabilities for

B R B R B R B R B B B R B B B R R R R B R B R B R B B B R B R

are exactly the same as

B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B B

These two events have exactly the same probability because each roll of the Roulette wheel is independent.

 

[Volkl]

So again, your one of those that believes that a human could in reality experience 50,000,000
black outcomes?

 

[micromass]

So again, your one of those that believes that a human could in reality experience 50,000,000
black outcomes?

The chance would be very small, but the chance is the same as any other outcome.

This is very easily checked with a computer. Just let a computer spew out random outcomes and see if there is any outcome that occurs more than another. Probability doesn't lie, my friend.

 

[Volkl]

I agree with the chance being small, so much so I'm betting on it. Please explain why you believe the chance is small? Do you believe the chance of ten in a row being black is much larger?

 

[micromass]

Whatever betting strategy you have, run it on a computer first. You'll see quickly if there are any fallacies.

 

[Volkl]

The logic here does not require a computer. The point is that there is a higher probability that smaller sets of like numbers occur than larger sets of like numbers, so there is a tendency for the next value to oppose the previous string of like values.

 

[micromass]

The logic here does not require a computer. The point is that there is a higher probability that smaller sets of like numbers occur than larger sets of like numbers, so there is a tendency for the next value to oppose the previous string of like values.

I know that seems believable to you, but the logic is inherently flawed. A value does not depend on previous values.

There is only one way to settle this and it is by practical experimentation. You will see in practical experiments that what you are saying is false.

 

[Volkl]

I guarantee that 1000 blacks values in a row would be found less than 100 black values in a row. Another way of saying it would be that there would be more sets of 100 blacks in a row then there would be 1000 black sets within the 50,000,000 sample size within all of the samples/trials any human could simulate in a lifetime.

 

[micromass]

I guarantee that 1000 blacks values in a row would be found less than 100 black values in a row.

Yes, of course. But this does not mean that if you have 100 black values in a row, that the chance on a red is somehow higher than the chance on a black...

 

[Kevin_Axion]

Micromass is right, there is only a variance in the probability if they are removed from the system.

 

[Volkl]

Since there is always more smaller sets of like numbers in a row then there is compared to larger sets of like numbers in a row - I believe it does. If not, you would not have said of course below.

 

[Volkl]

This system is simply 50,000,000 spins of a wheel though-Kevin.

 

[micromass]

This conversation is pointless. We can never convince each other in any possible way. So I suggest you go and try some practical experiments. Do it on a computer or in a casino. You'll see that what you say is false.

 

[Volkl]

How can the probability for 1000 blacks in a row be less than a hundred in a row if there was no tendency towards randomness?

 

[micromass]

How can the probability for 1000 blacks in a row be less than a hundred in a row if there was no tendency towards randomness?

I don't see what you're getting at here. The system is random.

 

[Volkl]

I'm trying to get agreement on the "tendency" towards randomness.

 

[Ignea_unda]

Each time you have a 50% chance of guessing right. There is ZERO impact from the choice before it.

 

[Kevin_Axion]

This system is simply 50,000,000 spins of a wheel though-Kevin.

Then there is no variance in probability. Each time you spin the wheel the probability of getting a number in your set \mathcal{S} is one over the cardinality of your set: \frac{1}{\left|\mathcal{S}\right|}.

 

[Volkl]

I'm trying to get agreement that there is a "tendency" towards randomness when comparing strings of like outcomes to other strings of like outcomes. I.e. Shorter strings of like outcomes are more prevalent then longer strings of like outcomes.

 

[micromass]

I'm trying to get agreement that there is a "tendency" towards randomness when comparing strings of like outcomes to other strings of like outcomes. I.e. Shorter strings of like outcomes are more prevalent then longer strings of like outcomes.

Nobody is debating here that a 1000 black outcomes are less prevalent than a 100 black outcomes. That is common sense.

However, what we're saying is that if the first 100 are black, then you have equal probability that the next one is red or black.

That is: the chance that you have 100 black and then 1 red is equal to the chance that you have 101 black.

 

[Volkl]

Micromass has already agreed to this concept that shorter strings of like outcomes are more prevalent then larger strings of like outcomes.

 

[micromass]

Micromass has already agreed to this concept that shorter strings of like outcomes are more prevalent then larger strings of like outcomes.

Not only that: shorter strings of a certain outcome (not necessarily like) are more prevalent than larger strings of a certain outcome.

That is: you will see more of the string

BRBRBRBBB

then of the string

BRBRBRBBBRRBRBBRBBRRRBBRB

So whether the outcomes are all black is irrelevant.

 

[Volkl]

If that is true then counting all the way up to a thousand following your same logic would mean that the 1000 has the same probability as the 100. Something is not right here and it has to do with the tendency for the string itself to be random as opposed to like valued I.e. All blacks.

 

[micromass]

If that is true then counting all the way up to a thousand following your same logic would mean that the 1000 has the same probability as the 100. Something is not right here and it has to do with the tendency for the string itself to be random as opposed to like valued I.e. All blacks.

I do not understand what you're saying here.

Anyway, I'm going to stop discussing this. It's pointless. Please try your theory out on a computer. You'll see that it's incorrect. And if it turns out you're right, do send the findings to me!

 

[paulfr]

I think the OP's fallacy is his mis-interpretation of The Law Of Large Numbers which says the experimental probability will match the theoretical probability ... but only " as n => infinity ".

Saying the next outcome will be Red after a long string of 100 Blacks is to make the approximation that 100 is almost infinity.

One does so at one's own peril

 

[Volkl]

The outcomes or the string of like values matters only as a way of showing that randomness affects strings of like values in a way that ultimately causes the gamblers fallacy to be false - because we agree on this concept of the tendency towards different values as compared to like values (strings of the same colour,like all black for instance).

 

[Volkl]

Paulf - your comments came across as oxymoronic to me, because the comment about the law of large numbers is exactly why I set a limit of 50,000,000, however you go on to believe that when we are not talking about infinity that the expected outcomes is somehow similar to the outcome as if we were talking about infinity.

 

[Volkl]

Each time you have a 50% chance of guessing right. There is ZERO impact from the choice before it.

Why are smaller sets of blacks more prevalent than larger sets of blacks then?

 

[Volkl]

If the Strings of blacks have different probabilities, but yet the individual spin outcomes have the same probability, what accounts for the strings of blacks having different probabilities?