Law of Large Numbers, and being "due".

BeerMoney · #1 09-13-2005, 05:37 PM

Our experiment will be that we flip a FAIR coin N times, and on one side it has -1 and on the other it has a 1. Let x be the sum of the N flips... Let Y be the average of those flips... Both E(x) = 0 , and E(Y) = 0.. That is, in the long run, we expect both to be zero..

Consider if we flip the coin 50 times and all 50 times it is a one! Now, X = 50, and Y = 1. Well, what if we flip the coin another 50 times.. Now, what is E(X) and E(Y)? For those next 50 flips, we still expect both the totaland the average to be zero.. So, When we finish those fifty flips, we still "expect" the total will be 50+0 =50, but we would expect the average to be (50+0)/100 = .5. Same logic, consider if we flipped 999,950 more times... We would expect the total to be x = 50 + 0= 50, and y = (50+0)/1,000,000= .00005.

Point: The average is going to zero AS A LIMIT. However, we don't expect to experience a -50 downswing at somepoint, just because we had a strang occurence of a +50 upswing.

I hope this made sense..

beta1607 · #2 09-13-2005, 05:41 PM

This is related to you losing for two months right?

Maybe you are not fliping the coin in a way to produce equal results?

Bartholow · #3 09-13-2005, 05:49 PM

What about it?

If this has to do with your downswing, it's true that you shouldn't expect that streaks will come to even each other out. On the other hand, with an infinite amount of time you should expect infinitely long streaks, as strange as that sounds. Or put another way, the best thing for you to do is figure out your observed win rate and standard dev, and use the math to figure out the A)likelihood that you are a winning player B) if you are a winning player, the likelihood that you'd have a streak of this length. It might require some stats work.

BeerMoney · #4 09-13-2005, 06:01 PM

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This is related to you losing for two months right?

[/ QUOTE ]

No, not at all. Where did I imply that?

The point is that if you lose, you lose, if you win, you win.. If you have a killer session, you shouldn't expect to get smoked in the next one, or vise versa. I see posts all over the place implying a notion of being due. MEbenhoe made a magazine article talking about earning theoretical $$. Two players with equal rates play for the same time, they both made the same theoretical $$, blah blah blah. No, one made money, the other lost.

With my example, we still expect our total to be 50 over the long hall.. That crazy streak of 50 never goes away, and we expect the total to still be fifty after 1 million trials, although the average will approach zero.

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Maybe you are not fliping the coin in a way to produce equal results?

[/ QUOTE ]

That wasn't the point. The example could have stated that you flip the coin once and observe a "1". The example still holds.

peritonlogon · #5 09-13-2005, 06:08 PM

[ QUOTE ]
On the other hand, with an infinite amount of time you should expect infinitely long streaks, as strange as that sounds.

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I hate this argument.... I hate it, I hate it, I hate it.
And I hate Cantor sets too. Like the number of all possitive integers is equal to all prime integers or, for that matter, all perfect integers. (this is not original) so if I start writing my autobiography, and it takes me it takes me a year to write about one day of my life assuming I live forever, at some point I'll finish it. Even though I think I'd get further and further from completion.

I'm trying to believe it, but I can't.

Bartholow · #6 09-13-2005, 07:27 PM

I'm not sure I follow you here. I'm somewhat familiar with Cantor sets, but I was really just trying to be practical and mention that streaks in poker can go beyond our ability to measure, but the likelihood decreases very quickly.

Andy B · #7 09-13-2005, 11:27 PM

Two points:

1) Huh?

2) Clearly, you do not drink enough.

Aicirt · #8 09-14-2005, 12:42 AM

Its not a limit. The probability that you get 1 fifty times in a row is the exact same as getting -1 fifty times in a row. While this highly unlikely (as is getting 1 fifty times in a row) given a large enough sample size you would expect to see a long string of -1 at some point. So yes, most likely it would take a long time before you got back to even, but it will happen given a large enough sample size. Its not like youll approach even and never get to it.

Aicirt

Kaeser · #9 09-14-2005, 01:00 AM

[ QUOTE ]

I hate this argument.... I hate it, I hate it, I hate it.
And I hate Cantor sets too. Like the number of all possitive integers is equal to all prime integers or, for that matter, all perfect integers. (this is not original) so if I start writing my autobiography, and it takes me it takes me a year to write about one day of my life assuming I live forever, at some point I'll finish it. Even though I think I'd get further and further from completion.

I'm trying to believe it, but I can't.

[/ QUOTE ]

This doesn't make any sense to me? In this scenario the gap between your work done and work remaining is continuously growing. In that case over an infinite time span wouldn't that gap approach infinity?

A related problem is the race between a rabbit and tortoise where the rabbit runs 10X as fast as the tortoise but the tortoise is given a 1000 yard head start. In the time it takes the rabbit to run 1000 yds. the tortoise has run 100. Then as the rabbit runs 100 the tortoise travels ten. In this case the gap is growing smaller.

#10 09-14-2005, 03:11 AM

[ QUOTE ]
This doesn't make any sense to me? In this scenario the gap between your work done and work remaining is continuously growing. In that case over an infinite time span wouldn't that gap approach infinity?

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There are EXACTLY as many days in an infinite period of time as there are years (sounds weird but it's true). So you would "finish" it at the "end" of inifinty. Except infinity never ends, so you never finish.