Math question with no real world significance

[Zetack]

In another forum a poster made a somewhat trollish statement that every poker player would go broke "eventually".

The reasoning being that, no matter how small your risk of ruin, given enough trials (an infinite number, if necessary) you will always hit your percentage chance of ruin, no matter how small.

Well that thought didn't interest me particularly, but it got me to thinking about math, and I since I suck at math, its very possible that I've broken my thinker and my head is now spewing out gobbeldygook. So let me run by my line of thinking, with an actual question at the end.

So I started with the idea that an improbable event becomes probable with enough trials. I.E. a million to one shot is pretty likely to occur if you run a hundred million trials.

so in relation to poker, if the deck is completely random, and if your odds of getting a specific pair, say 99, are 220-1. Then you odds of getting that specific pair back to back would be, what, 48,400-1? Pretty long, but if you play enough poker you're going to see it from time to time (heck I was dealt KK back to back in a live tournament on Sunday.)

And then your odds of getting your 99 three times in a row would be somewhere in the vacinity of 10 million to one. Ok, I'm not likely to see that, put you could envision poker playing computers getting thousands of hands per second hitting these odds.

So if you keep running the streak further out, pocket nines ten times in a row, 20, 1000... no matter how low the probability, if you ran enough trials eveuntually you would get a streak that long. Right?

And if you have an infininte number of trials, it doesn't matter how long or unlikely the streak is, eventually you will hit it. Given an infinite number of trials eventually you would have a streak of say pocket nines a million times in a row.

And it doesn't matter how big the number. A million times in a row, a billion, a quadrillion, evenutally you will hit the streak.

Now assuming the above is correct (is it, anybody?) Then I started to wonder, and here's where my head starts to hurt, given an infinite number of trials, will you eventually get a streak of pocket nines that itself goes on for an infinite number of draws?

[jay_shark]

lim x^n =0 where n goes to infinity and 0<x<1

So in other words , no .

[pzhon]

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In another forum a poster made a somewhat trollish statement that every poker player would go broke "eventually".

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This is wrong. A winning player will have arbitrarily large downswings eventually. However, by the time you experience a mammoth 2000 BB downswing, it is extremely likely that you have 20,000 BB or more in the bank, and so you don't bust out. Bankrolls don't have to start larger than the largest downswing you are likely to encounter.

The ROR if you are a 2/3 favorite on an even money wager is 1/2^bankroll, which is not 1.

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The reasoning being that, no matter how small your risk of ruin, given enough trials (an infinite number, if necessary) you will always hit your percentage chance of ruin, no matter how small.

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This is inconsistent with the normal meaning of ROR. If your risk of ruin is 1%, that means you do not go bust eventually 99/100 of the time. This contradicts the idea that you must bust out eventually.

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So if you keep running the streak further out, pocket nines ten times in a row, 20, 1000... no matter how low the probability, if you ran enough trials eveuntually you would get a streak that long. Right?


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Right, with probability 1.

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given an infinite number of trials, will you eventually get a streak of pocket nines that itself goes on for an infinite number of draws?

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No. There is a huge difference between a finite streak and an infinite streak. With probability 1, you are dealt infinitely many 72o hands. This is compatible with seeing arbitrarily long finite streaks of 99, but incompatible with getting nothing but 99 after some point.

[rufus]

This is a bankroll management issue. Using appropriate techniques and discipline, the risk of ruin for a profitable player can be made arbitrarily low.

[Phone Booth]

[ QUOTE ]
In another forum a poster made a somewhat trollish statement that every poker player would go broke "eventually".

The reasoning being that, no matter how small your risk of ruin, given enough trials (an infinite number, if necessary) you will always hit your percentage chance of ruin, no matter how small.

Well that thought didn't interest me particularly, but it got me to thinking about math, and I since I suck at math, its very possible that I've broken my thinker and my head is now spewing out gobbeldygook. So let me run by my line of thinking, with an actual question at the end.

So I started with the idea that an improbable event becomes probable with enough trials. I.E. a million to one shot is pretty likely to occur if you run a hundred million trials.

so in relation to poker, if the deck is completely random, and if your odds of getting a specific pair, say 99, are 220-1. Then you odds of getting that specific pair back to back would be, what, 48,400-1? Pretty long, but if you play enough poker you're going to see it from time to time (heck I was dealt KK back to back in a live tournament on Sunday.)

And then your odds of getting your 99 three times in a row would be somewhere in the vacinity of 10 million to one. Ok, I'm not likely to see that, put you could envision poker playing computers getting thousands of hands per second hitting these odds.

So if you keep running the streak further out, pocket nines ten times in a row, 20, 1000... no matter how low the probability, if you ran enough trials eveuntually you would get a streak that long. Right?

And if you have an infininte number of trials, it doesn't matter how long or unlikely the streak is, eventually you will hit it. Given an infinite number of trials eventually you would have a streak of say pocket nines a million times in a row.

And it doesn't matter how big the number. A million times in a row, a billion, a quadrillion, evenutally you will hit the streak.

Now assuming the above is correct (is it, anybody?) Then I started to wonder, and here's where my head starts to hurt, given an infinite number of trials, will you eventually get a streak of pocket nines that itself goes on for an infinite number of draws?

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Define infinite.

[pococurante]

The concept of infinity is retarded. It does not produce answers that make any sense, it's just theoretical BS.

For instance, it's mathematically true that .999999 repeating equals 1 (many long stupid arguments have been made, just trust me, according to the rules it does equal 1). Despite the fact that it "should" be less than 1 since it never actually reaches 1, the distance between .9999999 and 1 is infinitely small enough so that there actually is no distance.

So you could probably say that with an infinite number of hands, it's long enough to contain any possibility. It's long enough to contain an infinite streak of pocket 9s, as well as infinite streaks of pocket aces and infinite streaks of everything else.

Infinity is just an excuse to make up impossible nonsense, but be able to prove it.

[pzhon]

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The concept of infinity is retarded. It does not produce answers that make any sense, it's just theoretical BS.

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You'll find that mathematicians are not retarded (in fact, we intimidate rocket engineers), and mathematicians have several distinct, very precise, and useful concepts which can be called infinity (ordinals, cardinals, topological compactifications, algeraic infinities, calculus, etc.). The results of logic are not necessarily intuitive, and it is rare for untrained intuition to be good enough to agree with the consequences of dealing with infinity, but infinity is far from BS. Several of the notions of infinity (such as those in calculus) were introduced in order to get sensible answers to concrete problems, and to determine which manipulations of symbols make sense, and which do not.

[ QUOTE ]

For instance, it's mathematically true that .999999 repeating equals 1 (many long stupid arguments have been made, just trust me, according to the rules it does equal 1). Despite the fact that it "should" be less than 1 since it never actually reaches 1,


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Real numbers don't move or reach for anything. Numbers are not the same as notation. It happens that some numbers have more convenient representations than others, and there may be multiple ways of representing numbers, like 0 and -0.

You could come up with an alternate system of numbers which are based on notation and which wriggle and reach and try. Good luck creating a system as consistent and meaningful as real numbers.

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So you could probably say that with an infinite number of hands, it's long enough to contain any possibility. It's long enough to contain an infinite streak of pocket 9s, as well as infinite streaks of pocket aces and infinite streaks of everything else.


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You could say that, but you would be wrong. If there is an infinite streak of 99 hands, then after some point, the rest of the hands are 99, and the same is true for an infinite streak of AA hands. These are mutually exclusive. The correct statement is that every finite sequence will occur with probability one.

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Infinity is just an excuse to make up impossible nonsense, but be able to prove it.

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Infinity can be counterintuitive, but do not confuse your difficulties with intrinsic confusion within the subject.

Proper uses of infinity and infinitesimals simplify models and answers. What is the risk of ruin if you flip an unfair coin an infinite number of times, winning $1 with probability 2/3, and losing $1 with probability 1/3? 1/2^bankroll, and the proof is easy. What is the probability of busting out if you flip that coin only 1,000,000 times, rather than infinitely often? A mess.

[tarheeljks]

[ QUOTE ]
The concept of infinity is retarded. It does not produce answers that make any sense, it's just theoretical BS.

For instance, it's mathematically true that .999999 repeating equals 1 (many long stupid arguments have been made, just trust me, according to the rules it does equal 1). Despite the fact that it "should" be less than 1 since it never actually reaches 1, the distance between .9999999 and 1 is infinitely small enough so that there actually is no distance.

So you could probably say that with an infinite number of hands, it's long enough to contain any possibility. It's long enough to contain an infinite streak of pocket 9s, as well as infinite streaks of pocket aces and infinite streaks of everything else.

Infinity is just an excuse to make up impossible nonsense, but be able to prove it.

[/ QUOTE ]

your ignorance is staggering.

edit: also great reply pzhon

[AaronBrown]

<< given an infinite number of trials, will you eventually get a streak of pocket nines that itself goes on for an infinite number of draws? >>

I'd say it differently than pzhon. English is not precise enough to make the answer clearly yes or no.

Pick any number N, no matter how big. If you deal 221^N + N - 1 hands, you will get, on average, one sequence of N 99's in a row. If you deal, say, ten times this many hands, you will be virtually certain of getting at least one sequence of N 99's in a row. So as long as you specify N and p first, I can tell you how many hands you have to deal to have probability p of getting a sequence of N 99's in a row, and that answer is always a finite number.

On the other hand, you can name any number M and any p, and I can give you a number N such that there is less than probability p of getting a sequence of N 99's in a row during M deals (in fact, I don't have to think very hard, I can set N = M + 1 for any p).

So the question comes down to whether you name the number in the sequence first or the number of deals first. There is no limit to the longest sequence of 99's in an infinite number of deals, for any number N you name there will be a longer streak, but whether that means there is "a streak of pocket nines that itself goes on for an infinite number of draws" is not a question with a well-defined answer.

[Dire]

The odds of an infinite run of pocket nines would be lim(1/220^x : x=>infinity) which is 0. So the answer to your question is no, there would never be an infinite sequence. Math people are too verbose.