Is the Gambler's Fallacy Really a Fallacy?

[Ryan_m_b]

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.

Volkl it's quite simple in practice. Let's imagine you had a fair coin. The chance of heads or tails is 1/2. If you flip it once more the chance is the same. The difference comes when looking at the combination you are getting. Take a look at this image

Each coin toss has the same probability because there are two outcomes with equal likeliness;

H

or

T

However by the second coin toss there are 4 different possible combinations we could have had

HH

or

HT

or

TH

or

TT

So the chance of getting any specific combination at this point is 1/4. Do you see? And to work out the possibility of all the combinations of the next coin toss all we have to do is times 1/4 by 1/2 which will give us 1/8. If you check that on the image you will see that it's right, by the 3rd coin toss there are 8 possible and equally probably combinations.

 

[HallsofIvy]

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

I wouldn't say a "tendency" to randomness. The whole point of a roulette wheel is to have randomness right from the start.

meaning that there is a tendency to have less of the same particular value coming up in a row.

Again, "tendency" is the wrong word . The probability of, say, "222222" is exactly the same as "128435". Of course, the probability of "222222" is far smaller than the probability of "anything other than "222222" or even "not all 6 the same result", just as the probability of "128435" is far smaller than the probability of "anything other than "128435".

To me, this logic proves that the gamblers fallacy is in itself a fallacy.

Please explain how that proves it. Are you clear on exactly what the "gambler's fallacy" is?

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.

Well, I presume you can show that the probability of 50,000,000 runs of the same thing is much less that 1 over the number of people on Earth so it is extremely unlikely- but the whole point of "random" is that it could happen.

(On practical note, if a roulette wheel started giving the same result time after time, I am sure the casino would take it out of action long before it got to 50,000,000!)

 

[Volkl]

Well in theory
Thank you for identifying the heart of the concept I am getting at I.e. Small stings of blacks in a row occur more frequently than large strings of blacks in a row.

Formally, your confusion is ascribing some forcing on the part of http://en.wikipedia.org/wiki/Regression_toward_the_mean#Other_statistical_phenomenon". That is a result of the unlikeliness of having a long string, not the cause.[/QUOTE]

How does this unlikeliness you speak of get implemented in reality? This is the dicotomy and why it is ultimately related to the gamblers fallacy because the way this unlikeliness is implemented is to change to red and ultimately this acts like an added force - to the 50/50 equation that everyone loves prooving but is leaving out - that changes the likelihood of a black coming next to less than 50% I.e. A slightly safer bet on red.

 

[Volkl]

In a set of 50,000,000 what percentage (of the groups of blacks in a row) would be one black in a row? What percentage would be two blacks in a row. What percentage would be three blacks in a row? What percentage would be four blacks in a row? What percentage would be five blacks in a row?

Would red's percentages look the same as blacks?

 

[Fuz]

In a set of 50,000,000 what percentage (of the groups of blacks in a row) would be one black in a row? What percentage would be two blacks in a row. What percentage would be three blacks in a row? What percentage would be four blacks in a row? What percentage would be five blacks in a row?

Would red's percentages look the same as blacks?

We havn't tried this experimentally, so how could we know? Also, why does it matter whether the string of outcomes are of the same color? The roulette table doesn't know or care whether black shows up or red shows up. The permutation of N consecutive outcomes has the exact same theoretical probability of "showing up" as a different permutation of N consecutive outcomes, as stated a few times before.

If I draw a number out of a hat with values ranging from 1 to 100,000,000,000, drawing the number "444,444,444" is just as "random" as drawing "349,912,955".

 

[Hurkyl]

I just generated 100 random numbers from 0 to 1, and I got the following sequence:

1110010110001111001111111010111110000111110001111010011011111111001101010011001110110101100100111101
​

I couldn't believe it! The odds of getting that sequence are simply astronomical -- roughly one chance in 1,000,000,000,000,000,000,000,000,000,000.

After I saw the 99th digit, I was certain the last one had to be a zero -- after all, it was just so incredibly unlikely it would have been a 1 to give me the above sequence. But there you go, it was a 1 anyways!

 

[Volkl]

I'll bet you that there are more cases of single blacks than cases of two blacks in a row. Would you take that bet?

 

[Fuz]

I'll bet you that there are more cases of single blacks than cases of two blacks in a row. Would you take that bet?

Yes, I'll take that bet, because that is equivalent to saying RBR will show up more often than simply BB.

 

[Volkl]

That sequence of a hundred, displays the tendency for the groups of consecutive like numbers to be small. Whatever the reason, it means that a change is not only inevitable, but most likely frequent. Thanks for displaying it!

 

[Volkl]

You took that bet even after the post of the 100 random numbers? -you loose - now pay up ;)

 

[Fuz]

You took that bet even after the post of the 100 random numbers? -you loose - now pay up ;)

Explain please...

 

[micromass]

You took that bet even after the post of the 100 random numbers? -you loose - now pay up ;)

If Black = 1, then I counted 55 non-isolated blacks and 8 isolated blacks. I think you need to pay.

 

[Volkl]

Yes, I'll take that bet, because that is equivalent to saying RBR will show up more often than simply BB.

Imagine a longer set? You would be even more incorrect as the length approaches infinity.

 

[micromass]

Imagine a longer set? You would be even more incorrect as the length approaches infinity.

Quite the contrary.

Please post proof of your statements: theoretical or experimental.

 

[Volkl]

If Black = 1, then I counted 55 non-isolated blacks and 8 isolated blacks. I think you need to pay.

Three blacks in a row does not count as two combinations of two blacks in a row I.e. Only two blacks in a row count as two blacks in a row.

 

[micromass]

Question:

Would you believe that the following sequence of numbers is random:

010101010101010101010101010101010101010101010101010101010101010101010101010101

Why (not)?

 

[Fuz]

Imagine a longer set? You would be even more incorrect as the length approaches infinity.

The probability of drawing a red marble, then a black marble, then a red marble out of a bag is 1/8. The probability of drawing 2 consecutive black marbles out of a bag is 1/4. I was taught this when I took pre-algebra in 6th grade.

 

[Volkl]

Do you think there are just as many sets of 100 blacks in a row as there are individual blacks? Do you really need a simulation for this?

 

[micromass]

Do you think there are just as many sets of 100 blacks in a row as there are individual blacks?

Nobody is claiming this.

 

[Volkl]

The same concept holds true when comparing to 99, or 98, or 97...or 2. Doesn't it?

 

[micromass]

Please answer this:

Question:

Would you believe that the following sequence of numbers is random:

010101010101010101010101010101010101010101010101010101010101010101010101010101

Why (not)?

 

[Volkl]

Sure it is likely to be random because the values change, values tend to like doing that when things are random.

 

[paulfr]

No one has mentioned the Binomial Theorem and the Bernoulli Trial Formula.
For any finite sequence with known probability, these laws govern the outcomes.
I don't have time today to give a detailed example, but maybe someone can pick up the ball here for me.

Probability is fascinating partly because one single word or one misinterpretation can completely change a problem solution from correct to incorrect.

And like all Fallacies, The Gambler's Fallacy "seems to be true" only because the language obscures the whole truth of the situation.

 

[micromass]

So you believe that a sequence of 500000000 alternating occurences of 01010101 could occur in nature?? Wow...

 

[Fuz]

Sure it is likely to be random because the values change, values tend to like doing that when things are random.

Why? That's not at all what true randomness is.

Go here: http://www.random.org/cgi-bin/randbyte?nbytes=20&format=b and hit refresh a bunch of times.

 

[paulfr]

Sure it is likely to be random because the values change, values tend to like doing that when things are random.

You can not seriously believe that !
Endless 0101010101 ... looks random to you ?

If someone took a deck of playing cards and turned them over one at a time and they came out Red Black Red Black all the way thru the deck, you would NOT think the deck was stacked ? You would believe that was a true random outcome ?
If so, how could you ever tell if a pair of dice or a roulette table was weighted ?

 

[Volkl]

The chances of the values all being alternating is the same as them all being alike. I already talked about this being basically impossible previously.

Let me ask you this, if you believe that 100 blacks in a row will be less prevalent than ten blacks in a row, then how does this come about? What is the rule and is it not implemented by alternating the colour sooner rather then later?

 

[Volkl]

This is in the context of the original question which is a sample size of 50,000,000.

 

[Fuz]

Because:

1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2

>

1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2 * 1/2

 

[Well_InTheory]

Let me ask you this, if you believe that 100 blacks in a row will be less prevalent than ten blacks in a row, then how does this come about? What is the rule and is it not implemented by alternating the colour sooner rather then later?

You're still arguing that 'regression to the mean' is enforced by some higher power.

i.e. Given that highly unlikely events are usually followed by less unlikely events, you are saying it is because 'nature demands it' while in truth it's just that the probability that two unlikely events will happen is lower than one unlikely and one likely.

You don't lose the lottery the week after a big win because 'nature demands it to make up the numbers', it's just that the probability for each win is low, and the probability for both wins in a row is even lower. You can do the numbers to prove it.

Independent results (such as red/black on a roulette wheel) are independent. EVERY spin has a 50/50 chance of each.

 

[Volkl]

We should be comparing the total number of groups of ten to the total number of groups of 100. Do you believe the total number of groups of 10 as related to 100 would be in the same ratio as the ratio of probabilities you displayed?

 

[Volkl]

Well in theory
I think it is a safe bet to say that the chances of 100 blacks in a row is less likely than 10 in a row in the same way as I believe 11 blacks in a row is less likely than ten.

 

[Volkl]

If you wait for ten blacks in a row and bet against black only in this situation I.e(after ten blacks in a row), you will win more than 50 percent of the time, because it is less likely to have 11 blacks in a row over time.

 

[Well_InTheory]

If you wait for ten blacks in a row and bet against black only in this situation I.e(after ten blacks in a row), you will win more than 50 percent of the time, because it is less likely to have 11 blacks in a row over time.

No. This statement is exactly the gambler's fallacy.

You will win exactly 50% of the time.

The chances of BBBBBBBBBBB is the same as BBBBBBBBBBR. Pick any specific 11-long sequence and it will have the same probability of any other 11-long sequence.

Turning your thought-experiment around, you could also say you should bet on black, because BBBBBBBBBBR is a very unlikely result. Or "if you wait for BRBRBRBRBR then you should bet on R because continuing that sequence is unlikely", which is just as false.

 

[Jocko Homo]

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.

This is the single most ridiculous thread I have ever seen on this forum...

There are many ways to think about the Gambler's fallacy. Here's a simple logical argument that shows that the Gambler's fallacy cannot be true:

Suppose you have a coin that has a 50% probability of flipping heads and that coin flipped heads n times in a row. What are the chances of flipping heads a subsequent time? If that probability is anything other than 50% then you are contradicting the initial supposition that you had a coin that "has a 50% probability of flipping heads," therefore the only logically consistent conclusion is that that is the true probability. QED