Poker Gods or Fallacy?

[PantsOnFire]

If you are analyzing 10000 coin flips then it only has one starting point, flip 1. To go into the middle of the sample and see what the +/- is at that point is meaningless when you are analyzing the entire sample. And I think the reason for this is that the order of data is irrelevent since they are not interdependent.

If you took the results of a set of say 10 different and separate 10000 coinflip sessions, then the talley at a point 4328 would be different in each case. As a matter of fact, if you took a million 10000 coinflip sets, the average talley at point 4328 would probably be close to zero.

[DWarrior]

[ QUOTE ]
The problem is this...

Pick 10,000 coin flips.

Say the first 20 I flip heads. Over the 10,000 flips I should break even. I cant say that I'm more likely to flip more tails now so that it averages out. Over the next 9980 flips, I'm still expected to break even, which means that at the end of 10,000 flips, I'm expected to be down 20.

This is a contradiction to what I said before, but I think its correct. Meaning, there is no "luck credit". At any given point, thats what you are expected to be at at N hands from now.

[/ QUOTE ]

It's pretty simple to compute a confidence interval of the normal # of wins after 10k flips. I'm too lazy to do it, but I'm pretty sure the actual probability of you winning exactly 5k flips is pretty low. You'll most likely end up either over or under.

[PantsOnFire]

[ QUOTE ]
Likewise, if the first 10 flips come up heads, which is not that statistically bizarre, then at that point, the expected outcome of the next 990 flips is 495:495 and the updated expected outcome for all 1000 flips is 505:495. If the first 100 flips are heads, the updated expected outcome is 550:450, and so on.

[/ QUOTE ]
We all agree that one coinflip is not related to the next. That is, if we get 20 heads in a row, it is still 50/50 for the next flip.

So if we conduct 1000 flips and check the results, the bell curve would be highest at a zero difference and fall off from there outwards.

However, since the each coinflip is an independent event, then the order of the coinflips is irrelevent. So picking the first 100 has no meaning as far as what value they are related to the final outcome. You could essentially pick any 100 out of the 1000 in order or randomly and come up with a wide variety of results for those 100. But you can't use that information in any statical way in assessing the overall 1000 flip experiment.

All you would be proving is that the variance within the 1000 flip data set is greater variance than the 1000 flip data set itself. This should be evident.

Another way of looking at this is if you look at the first 100 flips and they are all tails, then you could statistically say that end result of the overall 1000 flips will more likely have 50 more tails than heads. However, is this meaningful? So we know 100 results and we don't know 900 thus we can make some calculations about now about the overall 1000 which are different than when we didn't know the results of a subset of that 1000. But is it correct to only "prelook" at the first 100? Why did we not look at the last 100 instead? or a random sample of 100?

I think the bottom line is that since each flip is independent, we can't label each flip as flip#1, flip#2, etc. And if we can't do that, then we can't say which flips happened "in a row". And if we can't say that, then we can't say we got 100 tails in a row. What we did was pick a random 100 out of the 1000 and happened to pick all tail results. This has no meaning.

[EvilSteve]

[ QUOTE ]
It's just mindboggling to me that, if you have 10,000 coin flips, you know the result should be about even. But, if you pick a starting point within that 10,000, say, like 4328 which has the talley at +32 heads or whatever, that you cant say that it will balance out.

[/ QUOTE ]
Looking at the tally difference brings up an interesting point I think. Because not only does this difference not approach zero, it actually tends to increase as the number of trials increases - that is, the absolute value of heads minus tails tends to increase. I'm referencing the Wikipedia article on "Random walk" here, since the quantity (heads minus tails) is the same as a one dimensional random walk from the origin, one step forward for heads, one step back for tails. Then distance from the origin is the absolute value of heads minus tails, and it turns out that average distance from the origin after n steps converges on square-root(2n/pi) which is about 0.8*square-root(n). I don't know the mathematical derivation of that (looked complicated, might involve the Gamma Function). But the point is, since the distance is increasing with the square root of n but not linearly, this doesn't prevent the percentage of heads from converging on 50%. Lets look at a sample where heads is always ahead of tails, and the difference (heads minus tails) is kept at 0.8 * sqrt(n). Yes that's clearly not random, but it shows how the tally differential can continue to increase while the ratios converge. I'll chart the following columns:
(1) number of trials = n
(2) heads - tails = 0.8 * sqrt(n) (rounded to even integer)
(3) heads tally
(4) tails tally
(5) percentage of flips that come up heads

n h-t h t %heads
10 2 6 4 60%
50 6 28 22 56%
100 8 54 46 54%
500 18 259 241 51.8%
1000 26 513 487 51.3%
5000 56 2523 2477 50.46%
10000 80 5040 4960 50.40%

The tallies keep moving further and further apart, but at a decreasing rate, and the percentage of heads is still converging on 50%.

[El_Hombre_Grande]

You can't say that your "luck credits" are going to pour in at any specific point, but you can expect that it is far more likely (on a % basis) to move towards even 50/50 in a larger sample than a smaller one. Unfortunately, because humans are finite beings that can only flip so many coins or see so many flops, we could croak before the Luck God graces us. So one can indeed be "lucky" or "unlucky" based on past results. The important point to take away is that past results NEVER NEVER NEVER have any bearing on the odds of anything particular happening on the next flip, which always remains 50/50. Thus you have the precise same chance on the next hand for yout QQ to hold up against Kx even though the boner next to you has flopped an K for the last umpteen hands, and has decimated your stack with his - EV play. The next hand is another universe, with the exact same odds as the last, and no interdependency whatsoever. You push him pre -flop not because he CAN'T POSSIBLY SUCKOUT AGAIN but rather because he is an idiot who is probably behind on the hand, again. There is a big difference in the two thought processes.

[CardSharkGames]

[ QUOTE ]
...if you look at the first 100 flips and they are all tails, then you could statistically say that end result of the overall 1000 flips will more likely have 50 more tails than heads. However, is this meaningful?


[/ QUOTE ]

Yes, it is meaningful. Those first 100 flips coming up tails are analogous to someone's Hold'em career starting off with 100 bad beats. Just as you no longer expect a 500:500 outcome in the coin flip trial, you can no longer expect that people starting with 100 bad beats will have, over their life times, an equal balance of good and bad luck.

[ QUOTE ]

So we know 100 results and we don't know 900 thus we can make some calculations about now about the overall 1000 which are different than when we didn't know the results of a subset of that 1000. But is it correct to only "prelook" at the first 100? Why did we not look at the last 100 instead? or a random sample of 100?


[/ QUOTE ]

Again, the analogy of the coin flipping trial is that someone has just started playing Hold'em and suffered 5 (or 10 or 100) bad beats in a row. This is not a "prelook", it is his actual experience to this point in time. What remains for such a player is the experiences he will have over the remainder of his life. Since the expected run of Hold'em luck for him from this time forward is 50:50 good:bad, it is obvious that such a person's initial bad luck is not statistically expected to "balance out" in the long run, as people like to claim.

Of course, all this assumes that card luck for any one individual is, indeed, subject to the laws of statistics. Doyle Brunson is known to argue otherwise (as is anyone who ever "plays the rush" after winning a pot or two).

[donkeykong2]

[ QUOTE ]


You wouldnt know it from this post, but I'm actually very good with statistics.

It's just mindboggling to me that, if you have 10,000 coin flips, you know the result should be about even. But, if you pick a starting point within that 10,000, say, like 4328 which has the talley at +32 heads or whatever, that you cant say that it will balance out.

Thats my problem. I'm still wrestling with it.

[/ QUOTE ]

your right, it doesnt balance out, just the head/coin relation will converge to 1 because the 20 more heads we have at some point will be less important with an increasing sample size ->100020/100000 just the extra flips done are expected to be even on average.
the variance of heads in absolute numbers increases with a greater sample size, while the variance of relative heads decreases.

[mvdgaag]

Past results have no influence on the present or future.
If you are conflipping and you are up $100 I expect you to have $100 on average anywhere in the future, given you have an infinite amount of money (if you don't you are bound to get unlucky and go broke somewhere in infinity).* There is noone keeping track of your luck or bad luck and you do not have any credits for getting lucky in the future.

GL

* for the smartasses, the next flip I expect him to either have $100 + bet or $100 - bet, he can't have $100 [img]/images/graemlins/tongue.gif[/img]

[AnyMouse]

[ QUOTE ]
* for the smartasses, the next flip I expect him to either have $100 + bet or $100 - bet, he can't have $100

[/ QUOTE ]

nh, sir.

[PantsOnFire]

[ QUOTE ]
Again, the analogy of the coin flipping trial is that someone has just started playing Hold'em and suffered 5 (or 10 or 100) bad beats in a row. This is not a "prelook", it is his actual experience to this point in time. What remains for such a player is the experiences he will have over the remainder of his life. Since the expected run of Hold'em luck for him from this time forward is 50:50 good:bad, it is obvious that such a person's initial bad luck is not statistically expected to "balance out" in the long run, as people like to claim.

[/ QUOTE ]
The problem with your reasoning is that you have picked an arbitrary subset out of the poker player's entire lifetime of hands. What is to stop me from going into his hand history and picking out a good run of 100 hands and saying now his expected remaining lifetime luck will be above 50% from that time on (and before that time for that matter)?

As another way of looking at it, let's say a guy plays 1 million hands in his poker career. If you take hands 1-100 and we see that he had a bad run and lost $1000 in those hands then we could say given equal luck he will be $1000 down over one million hands. But we could also go into hands 5000-5100 and see he had some very good luck and was up $1500. Now we would say with even luck for the other 999,900 hands, he will be up $1500 over his career.

So where is this going? Well if you picked all of the possible 100 hand subsets, some would be up and some would be down but in the end, they would average out to even over the million hands. In other words, you can't just pick out a random 100 hands and say since luck is even in the other 999,900 hands, then he will end up being up or down for amount won or lost for those 100 hands.

That's kinda like picking a good year for grapes and wine-making and saying since all other years will even out, then wine-making will be good in the long run.

So in the original argument, you picked out the first 100 hands of a large set of hands and tried to apply the statistical variance of those hands and apply it to the overall outcome of the total number of hands. You can't do that in statistics. And since one hand is independent of another, the order they occur is irrelevent. So when you analyse a lifetime of a poker players hands, they are a group without order, without a timestamp indicated when they happened because the "when" is not in the statistical equations.

So I guess when addressing the OP original question, it is fair to say that "luck" will be fairly even over a poker career. And it is equally fair to say that during that course, the current "luck" factor will vary from bad to good and back again. However, it is meaningless to pick out a particular set of hands where luck has been "bad" and try and determine what has or will happen in all the other hands. They are not dependent. You are looking at a sample of the overall set and any value given to that is meaningless since it is a statistical variation.

My first post was tongue in cheek. You cannot say after a bad start to a poker career that "luck" will even out that initial loss. To that I agree. However, the question itself is not posed in a way that is statistically meaningful so this is really a philosophical question and not a statistics question.