RANDOM thoughts
 

"My major hobby is teasing people who take themselves & the quality of their knowledge too seriously & those who don’t have the guts to sometimes say: I don’t know...." (You may not be able to change the world but can at least get some entertainment & make a living out of the epistemic arrogance of the human race). -- Nassim Nicholas Taleb

Nassim Nicholas Taleb is the author of a book called Fooled by Randomness, which I think is a must-read for any professional or aspiring professinal poker player. I read the book several years ago, but I think about it often as I continue to struggle to wrap my mind around the role of randomness in my results.

I am frequently reminded about the concepts explained in that book when I read posts on this forum, but lately it seems that randomness has become a particularly neglected concept. I think that understanding the role of randomness in the world is a key component to a sound world view and a happy, peaceful existence. Most people disagree with me.

The majority of people on this planet believe in ghosts. They believe that aliens pilot UFOs. They believe in their God at the exculusion of all other gods and believe that disagreeing with them may have dire consequences. They believe that everything happens for a reason. They believe in fate. They believe in destiny. Believing these things brings comfort to the majority of people on this planet... as ironic as it seems.

Now, there is another point of view regarding the laws of the universe. It is a point of view governed by science and reason. It is a minority viewpoint. It may or may not be any better or worse than the majority viewpoint, but it's easier to defend in a debate (using science and reason as the primary debate tactics, of course.)

What does this have to do with SNG poker? Well; smart, scientific, reasonable people get fooled by randomness all of the time. I'll let anyone interested read the book if they want some brilliant examples (including a nice discussion of the Monte Carlo simulator which made a brief but magnificent appearance on this forum a few months back.) In any event, I submit that we could all benefit from looking a little harder at the role of randomness in our results.

Somebody (read: practically everybody) will misinterpret this post as an assertion by me that so and so isn't as good as he thinks he is. This is not my point. I don't care how good so and so is. I just think that a concept as profoundly significant as randomness should make a more frequent appearance. An ample acknowledgement of randomness keeps egotism and idolatry in check. 5 billion people on this planet earn less than $10 per day. We need to be careful not to assume that the reason why we earn $10 per SNG is because we are smarter or better than the rest of the people on this planet. We are merely better served by randomness, that is all.

That doesn't mean that there isn't a correct way to play poker. There is. That doesn't mean that you can't become a winning poker player and make a lot of money. You can. Here's what it means:

Let's say that flipping coins became the next craze. MGM-Mirage decided to pay anyone that wanted to play even money if they could flip heads.(minus a fee, of course.) People love to do this, and pretty soon everybody is wagering everything they have trying to flip heads.

Let's say that a small percentage of people actually figure out how to use certain mechanics and weather conditions to improve the chances that they can flip heads. This select group of "professional flippers" averages heads 60% of the time. Some members of this elite group average more, some less. It's quite a hard skill to master... but worth the effort. But the rest of the public simply takes their 50-50 shot and pays the vig. Still, everybody is aware of this elite group, and many fancy themselves a member of this group... many more than actually belong.

So, eventually there's a group of 500,000 donkeys flipping coins and 25,000 professionals. Legend and lore builds, and the goal of every flipper is to make a "magic 9." A "magic 9" is 9 heads in a row. Only professionals are really capable of such a feat, legend has it, because the random odds of this happening are astronomical.

So everybody starts flipping. In the first 9 flips there will be close to 1000 "magic 9ers" from the random population and only 250 or so from the professional group. Clearly, anyone who makes a magic 9 will fancy themselves a professional, so there will be 1250 self-proclaimed professionals with the title of "magic 9er," while only 250 of those people are, in fact, professionals. Oh, and by the way, those 250 professionals are on quite a heater.

So, you can tweak the analogy anyway you'd like... but the bottom line is that as long as professionals comprise a small percentage of the total population of players in an endeavor where chance plays a role, the majority of successful individuals are going to be the beneficiaries of randomness. Furthermore, the most successful of the "skilled professional" group will also be beneficiaries of randomness to a much larger degree than they would like to admit.

My opinion is that by coming to terms with the statistical truism analogized above, it becomes easier to distinguish signal from noise in discussions about what to expect from the game of poker.

Or something.

Irieguy

Re: RANDOM thoughts
 

Many pokerplayers who do well do indeed have their magic 9 start. That gives them breething room to suck while learning to play properly since they now have a bankroll. For some that is not enough (que complaints about bad luck, cheating) to get them through. Most who quit poker never have their magic 9 at any point. It is just a stupid game where you lose money. And lastly we have those that have a bad start but since the signs that this thing can be beat are there keep trying and improving. They most likely will have their magic 9 sooner or later.

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But don't we talk about variance every day? I think most, or many at least, of us have a good idea of the role that variance plays in our results.

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But don't we talk about variance every day? I think most, or many at least, of us have a good idea of the role that variance plays in our results.

[/ QUOTE ]

If that was true, why is invariably most new posters talking about their ROI in their first post.

And why do the posters with experience only divulge their results after running good?

Re: RANDOM thoughts
 

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But don't we talk about variance every day? I think most, or many at least, of us have a good idea of the role that variance plays in our results.

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Yeah, like maybe 5 people on this forum have any idea of the depth and profundity that is VARIANCE.

I'm serious btw. Oh and Giga is one of them I'd imagine. He's rolled like 9 magic 9s, [img]/images/graemlins/tongue.gif[/img].

Yugoslav

Re: RANDOM thoughts
 

[ QUOTE ]
But don't we talk about variance every day? I think most, or many at least, of us have a good idea of the role that variance plays in our results.

[/ QUOTE ]

I think that it is almost like a buzzword for a lot of people. Someone post stats saying "I have a 50% ROI after 150 tournaments". Then well all say "Just wait for variance to kick in".

Meanwhile, the same poster who just posted the above statement about variance makes a seperate post saying "Look how well I did this month". His results are more reasonable and seemingly more attainable so we all say "Way to go" and assume that we should all aspire to have a 25% ROI at the 109s, or a 33% ROI at the 11s or whatever.

Meanwhile, the poster never posts the results of how well he did the other 11 months out of the year.

Re: RANDOM thoughts
 

Well, talking about your ROI doesn't mean you have no concept of variance. If you played 50 tournaments and you had an ROI of 50%, well, you had an ROI of 50%.

However, I'm not going to speak for, or defend, the new posters who come in here thinking they are 70% ITM, etc. I'm just speaking for the majority of the people who have read the forum for more than a day.

Basically, I'm just saying, did that big long post say something besides "variance variance you are running good blah", because if so, I'd like to be enlightened.

This isn't meant as a dig, as I've gotten plenty of insight from Irie's posts before. I am just trying to make sure there isn't some new nugget of knowledge buried in this one.

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You dont need to roll a magic 9 if you start with play chips and work your way up to .01-.02 nl and so on. I don't understand why everyone doesn't do that.

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I think that it is almost like a buzzword for a lot of people. Someone post stats saying "I have a 50% ROI after 150 tournaments". Then well all say "Just wait for variance to kick in".

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Also it is used to explain away losing streaks. Drop xx buyins. "It is just variance". Which is silly. For all we know it could have zero to do with variance and everything to do with poor play.

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You dont need to roll a magic 9 if you start with play chips and work your way up to .01-.02 nl and so on. I don't understand why everyone doesn't do that.

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Hehe, I always get confused by those that read three 2+2 books while playing with play money and then make their first deposit.

Those freaks scare me.

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fine be that way. [img]/images/graemlins/tongue.gif[/img]

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[ QUOTE ]
They believe in their God at the exculusion of all other gods and believe that disagreeing with them may have dire consequences. They believe that everything happens for a reason. They believe in fate. They believe in destiny. Believing these things brings comfort to the majority of people on this planet... as ironic as it seems.

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I am comforted by these things and I wonder, if as a doctor you have ever experienced anything that cannot be explained by simple randomness.

Re: RANDOM thoughts
 

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I am comforted by these things and I wonder, if as a doctor you have ever experienced anything that cannot be explained by simple randomness.

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Randomness is far from simple.

Irieguy

Re: RANDOM thoughts
 

[ QUOTE ]
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I am comforted by these things and I wonder, if as a doctor you have ever experienced anything that cannot be explained by simple randomness.

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Randomness is far from simple.

Irieguy

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What isn't simple about it? Or do I just need some puffs on the peace pipe to go with my alcohol tonight...

Re: RANDOM thoughts
 

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Somebody (read: practically everybody) will misinterpret this post as an assertion by me that so and so isn't as good as he thinks he is.

[/ QUOTE ]

You wrote this whole essay just to say how much I suck. Nice.

Re: RANDOM thoughts
 

I certainly agree with the OP that the 2+2 approach would benefit from more science and less pride and/or emotion.

Part of applying a scientific approach to poker is to understand variance and it's roll. But variance doesn't have to be a parlor word around here. We can study the extent to which randomness can explain events. Statistics gives us a boatload of tools to help identify patterns in seemingly random data.

Yes, up and down swings can happen by random chance. However, if my data shows that I have 40% ROI over 100 tournaments, that is enough to be about 99% confident that I am a winning poker player if I continue to utilize the same strategies. Similarly, if I have 15% ROI over 10,000 tournaments but I've run at -20% ROI for the past 100, the probability is high that I've been playing poorly and I need to revisit my strategy.

2+2 should spend more time understanding and modeling the random walks of poker careers.

One easy model is to use one-sided hypothesis testing to predict the lowest possible ROI a given player can expect long run based on their current sample.* A second useful tool is confidence intervals for players with a huge database of tournaments to help them detect when they might be playing badly.

* Of course, we can only measure this to a specified degree of confidence; we can never be 100% sure we aren't just lucky.

Re: RANDOM thoughts
 

Nice post.

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Newton said for every action there is a reaction. He broke things down into a measurable, predictable, mechanical science. Individualism and Western Society developed out of these sorts of ideas (or something). We're comfortable with these concepts.

Screw Quantum Physics. Heisenberg is full of sh-t, God does not play dice (One of greatest f-ing quotes of all-time BTW). If I average 6 BB/100 for 2,000 hands of limit poker or a 40% ITM for 100 higher level SNGs, I am a genius brilliant player.

If I go on a horrid losing streak, internet poker is rigged, and when I go another another heater it is because the "Party rigging switch" has been flipped back to off, and everything is right in the universe...

Re: RANDOM thoughts
 

Weird.

If you're not trying to say that some results are based on luck, then what are you trying to say?

I understand your point in general, but it's relation to SNG poker is slim at best, complete gibberish at it's worst.

Whatever.

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Seriously, interesting post. I'll check out the book, although he's probably preaching to the choir with me.

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read the mid/high limit DERB thread lately?

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[ QUOTE ]

I understand your point in general, but it's relation to SNG poker is slim at best, complete gibberish at it's worst.

Whatever.

[/ QUOTE ]

Re: RANDOM thoughts
 

hey guys interesting post. i have been running very good with a 25% ITM and 10% ROI over 95 games.. i think this is good but is it enuf to tell if im a winning player or not?




AHHHHAAAAHHAAAAAAHAAAAA *do the raptor **GASP** * AHHHHAHHAAAHHHHHHHAAAAA

YOU CAN BE MY BABY IT DONT MATTER IF YOUR BLACK OF WHITE

Re: RANDOM thoughts
 

I think it was (or at least could become) a more constructive response to "Part-time SnG grinders" (thread link).

I mean even ignoring the amazing drjnightowl there were several posters who have a hard time understanding randomness.

And that isn't surprising seeing people in general have a hard time with probability. And not just random calculations (What are the chances of the flopped four flush giving me the flush be the river?) but especially the intuition to reason properly with math especially under uncertainty (and I think it is funny that Irieguy mentions "Fooled by Randomness" as that is on my queue of books to read and I'll probably get to it after HOH2 and HP6). I mean I post a little in General Gambling/Probability and look at two common questions people ask:

- Why doesn't a Martingale roulette strategy work (bet $1 on black, if you lose double your bet until you win and start over again)?

- The Monty Hall/Let's make a deal problem (choose from 3 doors randomly (one of which has a prize behind it), after you pick the host (who knows what's behind what door) will show you one of the rooms that you didn't pick that has no prize and then offer you to keep your pick or switch to the unpicked and unopened room).

The first is obviously a problem understanding how fast exponential growth is and figuring out how to calculate the expectation in the face of that. And what's worth even when people can't calculate the expectation they can't even inuitively reason why it doesn't work.

The second is but one of many examples that most people just don't get conditional probability and also can't logically reason the answer very well. I have two favorite examples of conditional probability that, when I ask them to many smart people, they can't figure out the right answer (or even worse "figure out" a wrong answer).

Question 1:
Assume we are playing a game where I will roll a pair of normal fair 6-sided dice. If no die shows a 6, then I will reroll both die. When I stop rolling the dice (because one or more die has a 6) I will pay you $X if both dice show 6. If instead there is a 1, 2, 3, 4, or 5 to go with the 6 when I stop rolling the dice you will pay me $1. What is the value of X that makes this a fair game? (If you were to offer X=8 to most people do you think they'd play the game?)

The second one of my questions (and I think I may be cribbing it from "There Are Two Errors in the the Title of This Book*: A Sourcebook of Philosophical Puzzles, Paradoxes and Problems" by Robert M. Martin which is also a fantastic read) ties directly in to what Irieguy was getting at with the coin-flipping analogy (although coin-flipping is a bad analogy because one really can "flip" coins in an unfair way as often coins don't flip but actually wobble which only looks like flipping and one can practice this). But first the question:

Question 2 (a different flavor of the false positive health test question):
Assume that we are on an Island with 100 taxis. On the Island 95 taxis are yellow and 5 taxis are green on the Island. Police know that eyewitnesses are 80% accurate (that is they will correctly identify the color of a taxi 80% of the time and the other 20% of the time they will say that the taxi was the opposite color from what it was - and that this eroor rate is idependent and unrelated to the actual color of taxi). There is a hit and run involving a taxi. The only eyewitness says that the taxi was green. What is the probability that the taxi involved was actually a green taxi?

The very high number of people who think that it doesn't matter if you switch in the Montey Hall problem because it is 50/50 that you'll win or that the answer to question 1 above is 5 or that the answer to question 2 is 80% is astonishing.

So to repharse Irieguy's analogy as it applies to SNG (and this is the important point):

So in SNG land imagine someone has done the heavy lifting of the math and figures out that the chances of a break-even player having a 15% (or better) ROI over 500 SNG is just 20%. Player X just finished 500 SNG and had a 15% ROI. Player X now thinks they are 80% likely to be a winning player. The biggest mistake here is that Player X isn't taking in to account the distribution of the population. The vast majority of SNG players are losing players (for the sake of argument let's say 95%). The average SNG player loses the rake. So player X isn't somehow picked from a uniform distribution that is as likely to be -100% ROI as +100% ROI but rather a distribution centered on -rake ROI that gets drastically less players the further you move from -rake ROI. This is now a conditional probability question just like question 2 above and actually player X is much more likely to be a break-even or losing player currently being misidentified by our "one witness" (the 500 SNG sample) then a true winning player being correctly identified.

Irieguy's secondary point:

"Furthermore, the most successful of the 'skilled professional' group will also be beneficiaries of randomness to a much larger degree than they would like to admit" is worth remembering too (and has Daniel Negreanu's name all over it, for one).

Re: RANDOM thoughts
 

[ QUOTE ]
hey guys interesting post. i have been running very good with a 25% ITM and 10% ROI over 95 games.. i think this is good but is it enuf to tell if im a winning player or not?




AHHHHAAAAHHAAAAAAHAAAAA *do the raptor **GASP** * AHHHHAHHAAAHHHHHHHAAAAA

YOU CAN BE MY BABY IT DONT MATTER IF YOUR BLACK OF WHITE

[/ QUOTE ]

Based on the stated sample, there's 70.5% confidence that you are ROI >= 0.

Re: RANDOM thoughts
 

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The very high number of people who think that it doesn't matter if you switch in the Montey Hall problem because it is 50/50 that you'll win . . . is astonishing.



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Oh yes, let's discuss Monty Hall.

Because it is 50/50. Marilyn vos Savant was wrong.

Re: RANDOM thoughts
 

I think this little essay is brilliant.

However, there is a small flaw in it.
There would be some real lucky flipdonkeys who performed a 'magic 9' yet wouldnt claim to be a pro.

Not that the real pros would notice.
A lucky poker player

Re: RANDOM thoughts
 

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I am just trying to make sure there isn't some new nugget of knowledge buried in this one.

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Irie doesn't dispense knowledge in nugget-sized parcels; it's more like "pixie dust" [img]/images/graemlins/smirk.gif[/img]

Re: RANDOM thoughts
 

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Because it is 50/50. Marilyn vos Savant was wrong.

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Given her assumptions, she was correct. it is NOT 50/50 if the host always offers the switch.

Regards
Brad S

Re: RANDOM thoughts
 

Very good post.

I actually doubt that many who should be paying attention to this will really get it. Bayesian reasoning is just so counter-intuitive to so many.

Regards
Brad S

Re: RANDOM thoughts
 

[ QUOTE ]
[ QUOTE ]
Because it is 50/50. Marilyn vos Savant was wrong.

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Given her assumptions, she was correct. it is NOT 50/50 if the host always offers the switch.

Regards
Brad S

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yep, but why did she make that assumption? Did she watch every episode of "let's make a deal?"

What if you assume that the producers of the show are cheap (probably a more reasonable assumption)? They only offer the switch when you choose the car. Then you are certain to lose the car if you switch.

Statistics should reflect the real world, not the other way around.

Marilyn vos Savant is an idiot with an IQ of 200. Not mutually exclusive things.

Re: RANDOM thoughts
 

I think that when you contrast the accepted opinion, as summarized by valejo:

[ QUOTE ]
One easy model is to use one-sided hypothesis testing to predict the lowest possible ROI a given player can expect long run based on their current sample.* A second useful tool is confidence intervals for players with a huge database of tournaments to help them detect when they might be playing badly.



[/ QUOTE ]

with what SumZero later says in this thread about bayesian implications on our usual confidence assumptions

[ QUOTE ]
So to repharse Irieguy's analogy as it applies to SNG (and this is the important point):

So in SNG land imagine someone has done the heavy lifting of the math and figures out that the chances of a break-even player having a 15% (or better) ROI over 500 SNG is just 20%. Player X just finished 500 SNG and had a 15% ROI. Player X now thinks they are 80% likely to be a winning player. The biggest mistake here is that Player X isn't taking in to account the distribution of the population. The vast majority of SNG players are losing players (for the sake of argument let's say 95%). The average SNG player loses the rake. So player X isn't somehow picked from a uniform distribution that is as likely to be -100% ROI as +100% ROI but rather a distribution centered on -rake ROI that gets drastically less players the further you move from -rake ROI. This is now a conditional probability question just like question 2 above and actually player X is much more likely to be a break-even or losing player currently being misidentified by our "one witness" (the 500 SNG sample) then a true winning player being correctly identified.

Irieguy's secondary point:

"Furthermore, the most successful of the 'skilled professional' group will also be beneficiaries of randomness to a much larger degree than they would like to admit" is worth remembering too (and has Daniel Negreanu's name all over it, for one).


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then yes, I think that something interesting is being said in Irie's post. Unfortunately, it seems to be being missed by many.

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I understand your point in general, but it's relation to SNG poker is slim at best, complete gibberish at it's worst.


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So no, I don't think you do understand. And I don't think it is gibberish. Though I suppose It's relation is no more specific to SNGs than to any other kind of poker.

Regards
Brad S

Re: RANDOM thoughts
 

I have a question now... especially seeing as I'm the one who first injected confidence interval calculations into this forum with my confidence calculator and later spreadsheets.

What effect does this have on our actual winning confidence? If I calculate a winning confidence of 95% based on the assumptionthat (say) 2/3 of all players are really losers, how does this sway the chance that I am a winner, given the additional info I now have about the rest of the population? What if 90% are losers?

I actually think I can answer this, but the question has just hit me, and you seem well versed.

How do I determine this new confidence % based on the rest of the population? (I may need to update the confidence calculator)

Regards
Brad S

Re: RANDOM thoughts
 

[ QUOTE ]

So in SNG land imagine someone has done the heavy lifting of the math and figures out that the chances of a break-even player having a 15% (or better) ROI over 500 SNG is just 20%. Player X just finished 500 SNG and had a 15% ROI. Player X now thinks they are 80% likely to be a winning player. The biggest mistake here is that Player X isn't taking in to account the distribution of the population. The vast majority of SNG players are losing players (for the sake of argument let's say 95%). The average SNG player loses the rake. So player X isn't somehow picked from a uniform distribution that is as likely to be -100% ROI as +100% ROI but rather a distribution centered on -rake ROI that gets drastically less players the further you move from -rake ROI. This is now a conditional probability question just like question 2 above and actually player X is much more likely to be a break-even or losing player currently being misidentified by our "one witness" (the 500 SNG sample) then a true winning player being correctly identified.

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That's interesting. I'll stick some numbers on it to make it clearer.

Suppose we have 1000 players. 950 of these players are losing players, and 50 players are winning players.

After 500 tournaments, 10% of the losing players have a positive ROI, and 100% of the winning players have a positive ROI. This means that 95 of the losing players have a positive ROI, and 50 of the winning players have a positive ROI. If you are one of the 1000 players and have a positive ROI, are you more likely to be a winning player or a losing player?

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Irieguy, you just have to kick over the ant hill every so often, don't you? [img]/images/graemlins/wink.gif[/img]

Your post leads me to something that I think about alot regarding my own circumstances:

It came up a while back when a player who had an ROI of like 8% on the 11s (after 500 games) was asking if he should move up. He wanted to know how other people did on the 11s. I answered that I have well over 1000 games on the 11s with an ROI over 30%. I was immediately criticized (Flyingmoose, I believe) and asked if I was the most "cashout happy person in the world". My resonse is that I do throw in the occassional 22 with good results and that soon I will move up to that level. But I want to get 1000 games in at each level (as a winner) before moving up. Wherever I struggle is where I will stop (and try to learn) and that my ego will not get in my way - if I hit the 55s some day and lose, I will move back down.

I made the point that if someone is only beating the 11s at an 8% rate, the last thing they should do is move up and multi-table. He should improve his game, learn, and wait until he can beat the 11s at a higher rate. My belief is tht if you simply play solid poker and let your opponents beat themselves (and know how to play the bubble, of course), anyone should be able to beat the 11s for (pick a number) 15% ROI at least.

So, basically, the "randomness" idea (which I agree with in general) loses some of its power at the lower levels, since the terrible play of your opponents is not random, but entirely predictable and expected. Their bad play alone is enough to make you a winner in the long run.

My point is that I find it interesting that most of us (myself included) agree with you in general about "randomness" (or variance or whatever else you wish to call it) but at the same time, I believe that so many people move up in limits too quickly. Many players routinely give what I think is dangerous advice to noobs advising them to do so. This is in spite of the fact that GOOD players regularly post horror stories of bad runs ruining their bankrolls after moving up.

Personally, I have no ego when it comes to poker. TONS of players are better than me and I have no problem admitting it. I have lots to learn. But in the meantime, I have no shame winning at the lower levels until I can determine that I am ready to move up.

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Surely the problem with this model is it makes the assumption "assuming you're one of the 1000 players". This isn't a fair assumption. If you're a regular here, read and post threads, read 2+2 material, study your HHs and really try and understand the game, you are not part of the general distribution of the 1000 players.

Infact, it could work in the opposite way: say 95% of 1000 2+2 posters (who spend a good amount of time trying to really learn and study the game) have an average ROI of 10% over 500 SNGs, yet your ROI is -5%, what's the chance you are a winning player?

Keep in mind that we have a famous thread here on 2+2 which lays out clearly and exactly how to play to 'beat the 10+1s'.

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Well the problem is these probability tests are for a 'random' player. Someone who just plays, wins and then posts X% after 100 sngs is likely just lucky. Someone who works on their game and achieves the same X% after 100 sngs is more likely to be a winning player than the former, but if you ran a test the numbers would be the same.

It's like when I was running tests for sportsbetting. If you seached for patterns and found something that was a 1 in 10k event, it's still meaningless. But if you have developed your hypothesis first and then tested and achieved the same results then it means a WHOLE lot more.

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<-- waiting and working on my "M9"

Re: RANDOM thoughts
 

[ QUOTE ]
So in SNG land imagine someone has done the heavy lifting of the math and figures out that the chances of a break-even player having a 15% (or better) ROI over 500 SNG is just 20%. Player X just finished 500 SNG and had a 15% ROI. Player X now thinks they are 80% likely to be a winning player. The biggest mistake here is that Player X isn't taking in to account the distribution of the population. The vast majority of SNG players are losing players (for the sake of argument let's say 95%). The average SNG player loses the rake. So player X isn't somehow picked from a uniform distribution that is as likely to be -100% ROI as +100% ROI but rather a distribution centered on -rake ROI that gets drastically less players the further you move from -rake ROI.

[/ QUOTE ]

I'm not a mathematician, but I think this isn't very correct. If you so insist on the importance of conditional probablity to understanding results in this game, you can't ignore "deeper levels" of conditional probability. For some reason, you seem to ignore them completely, IMO.

If you state in the beginning of this example, that we're talking about a player who "has done the heavy lifting of the math" involved here, it's only natural to assume this player is not picked up randomly from the non-uniform distribution of SNG players, but rather that he is placed in a "higher" point on this graph to begin with, AT LEAST with regard to his mathematical abilities, which, as we know, play a very important part in playing SNGs. And so, using some conditional probability reasoning, it makes sense to conclude that there IS something along the lines of 80% confidence (or even more!) that this is indeed a winning player.

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[ QUOTE ]

Question 1:
Assume we are playing a game where I will roll a pair of normal fair 6-sided dice. If no die shows a 6, then I will reroll both die. When I stop rolling the dice (because one or more die has a 6) I will pay you $X if both dice show 6. If instead there is a 1, 2, 3, 4, or 5 to go with the 6 when I stop rolling the dice you will pay me $1. What is the value of X that makes this a fair game? (If you were to offer X=8 to most people do you think they'd play the game?)


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I believe that x=5 is a fair game (on average both people break even). 6:1-6:5 = 5 ways, 6:6=1 way any other outcome is irrelevent...

maybe I should have finished reading thread before posting, but if it is not 5, then what is the answer? Certainly over the short term you may rarely or frequently see 6:6, but over long term x should = 5, if not please explain why not.