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 05-20-2012, 11:08 AM #1 centurion   Join Date: Aug 2011 Posts: 121 So Many Screw Up Law of Large Numbers True or false, I keep hearing statements like "Your dollar value of wins at gambling will approach your expected wins in the long-run" Is this an accurate statement? Why or why not?
 05-20-2012, 02:21 PM #2 stranger   Join Date: May 2012 Posts: 3 Re: So Many Screw Up Law of Large Numbers The ratio between the two will approach 1 in the infinite run.
 05-20-2012, 02:49 PM #3 Carpal \'Tunnel     Join Date: Jun 2005 Location: Psychology Department Posts: 7,430 Re: So Many Screw Up Law of Large Numbers I would say it is not accurate. Your return on investment percentage (or ratio as the other poster said) approaches your expected rate of return as your number of investments (N) increases. However, the raw total won/lost will vary randomly around expectation from its current point.
05-20-2012, 03:21 PM   #4
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Re: So Many Screw Up Law of Large Numbers

Quote:
 Originally Posted by random_person True or false, I keep hearing statements like "Your dollar value of wins at gambling will approach your expected wins in the long-run" Is this an accurate statement? Why or why not?
That statement is absolutely false.

The average absolute difference in dollars between results and expectation gets larger as trial sizes increase. The ratio of the difference to the expectation is what gets smaller. This is simply a result of the denominator growing faster than the numerator. It's a square root relationship, but BOTH get larger.

The law of large numbers applies to the ratio, not to the absolute dollar difference and not to "variance". Variance as a quantity gets larger as trials increase, by definition and in practice.

For more detailed explanations see the Probability forum, this has been discussed a hundred times.

Last edited by NewOldGuy; 05-20-2012 at 03:41 PM.

05-20-2012, 03:56 PM   #5
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Re: So Many Screw Up Law of Large Numbers

Quote:
 Originally Posted by NewOldGuy That statement is absolutely false The average absolute difference in dollars between results and expectation gets larger as trial sizes increase.
Correct.

Quote:
 The ratio of the difference to the expectation is what gets smaller. This is simply a result of the denominator growing faster than the numerator. It's a square root relationship, but BOTH get larger.
You mean the ratio of the difference to the total number of hands or to the total amount bet, not to the expectation.

Your win rate in bb/100 hands approaches your true EV in bb/100 hands (assuming that the later doesn't vary in a way that makes it impossible to define).

Quote:
 Originally Posted by 10000MegaWat The ratio between the two will approach 1 in the infinite run.
That would mean that they approach each other as the original statement said, and the original statement is false. EDIT: This is clearly not true, and 10000MegaWat's statement is OK and not the same as the original false statement.

Last edited by BruceZ; 05-21-2012 at 12:20 PM.

05-20-2012, 03:57 PM   #6
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Re: So Many Screw Up Law of Large Numbers

Quote:
 Originally Posted by BruceZ You mean the ratio of the difference to the number of hands or to the total amount bet, not to the expectation.
Yes, to the total, thanks.

05-20-2012, 11:57 PM   #7
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Re: So Many Screw Up Law of Large Numbers

Quote:
 Originally Posted by 10000MegaWAT The ratio between the two will approach 1 in the infinite run.
No, it won't. This is exactly the point being made in the title.

05-21-2012, 11:27 AM   #8
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Re: So Many Screw Up Law of Large Numbers

Quote:
 Originally Posted by BruceZ That would mean that they approach each other as the original statement said, and the original statement is false.
No, that wont. For ex., , yet x+1 and x don't approach each other.

Also, "Your win rate in bb/100 hands approaches your true EV in bb/100 hands" is a different formulation of what i said. The ratio between win rate and true EV is the same, whether you measure both in bb/100, just bb, or \$.

Quote:
 Originally Posted by NewOldGuy No, it won't. This is exactly the point being made in the title.
Same story as above... You agree with me and then say that i'm wrong. I say a/b -> 1, you say |a-b|/a -> 0. Same ****.

05-21-2012, 11:54 AM   #9
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Re: So Many Screw Up Law of Large Numbers

Quote:
 Originally Posted by 10000MegaWAT Same story as above... You agree with me and then say that i'm wrong. I say a/b -> 1, you say |a-b|/a -> 0. Same ****.
The average absolute difference in dollars between the result and the expected result, gets larger as sample sizes increase. It grows at at a rate of the square root of the sample size increase. The ratio of the two does not approach any number, much less 1.

05-21-2012, 12:04 PM   #10
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Re: So Many Screw Up Law of Large Numbers

Quote:
 Originally Posted by NewOldGuy The average absolute difference in dollars between the result and the expected result, gets larger as sample sizes increase. It grows at at a rate of the square root of the sample size increase. The ratio of the two does not approach any number, much less 1.
oh yes it does

edit: pic for the lazy.

05-21-2012, 12:15 PM   #11
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Re: So Many Screw Up Law of Large Numbers

Quote:
 Originally Posted by 10000MegaWAT No, that wont. For ex., , yet x+1 and x don't approach each other.
Oops, that was dumb of me. You're statement was OK, and not the same as the original.

 05-22-2012, 01:31 PM #12 Pooh-Bah     Join Date: Sep 2004 Posts: 4,329 Re: So Many Screw Up Law of Large Numbers Does the ratio of the result to the expected result approach 1 for a series of 0ev coinflips?
 05-23-2012, 12:27 PM #13 centurion   Join Date: Aug 2011 Posts: 121 Re: So Many Screw Up Law of Large Numbers I think its best to think about it as averages converging to expectations. Like the probability that Xbar-X is greater than any specific small number converges to zero. The ratio version obviously has a problem if the denominator is zero. Anyway, I'm happy everyone here gets the point. Xbar converges to X because the denominator of the average gets large at a faster rate than the numerator. However, your gambling wins and losses in dollar terms really matter in the sense that future gambles are independent of the past. I hear people act like short term losses should be ignored. That's fine from an emotional standpoint because you don't want losses to take a psychological toll, but the way the point is explained often messes up the math.

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