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09-07-2005, 09:26 AM
Is it true in probability theory that "anything that can happen will happen"?

Say you're playing Roulette even chances (red/black), the chance of red appearing in a spin is 47.368%.

The chance of getting 10 reds in a row is 0.0568% (0.47368 to the power of 10). The chance of not getting 10 reds in a row is 99.94%.

0.0568/100 x 2000 = 1.136
Therefore, in the long run around 1 string of 10 reds will appear in 2000 rounds of 10 spins each. (sorry martingale players hehe)

Is this correct? Since the probability cannot be reduced 0%, even with 0.0000276% chance of getting 20 reds in a row we will and do get 20 reds in a row. Does it mean that there is even a possibility of 100 reds in a row? Or even 1000 reds in a row? /images/graemlins/shocked.gif

LetYouDown
09-07-2005, 10:22 AM
Yes. Granted, for some things (100 red in a row), you'll need a huge timeline/# of spins, but it will happen on an infinite timescale. The odds of getting 100 red in a row in the span of a human life, spinning constantly every 30 seconds are still very small...but I assume you're talking about an infinite timescale.

09-08-2005, 12:02 AM
[ QUOTE ]
Does it mean that there is even a possibility of 100 reds in a row? Or even 1000 reds in a row? /images/graemlins/shocked.gif

[/ QUOTE ]

Yes, or 10000, or 100000... etc.

It gets worse: there is as much chance to get such a sequence, beginning from your first observation, as any other specific sequence you may choose.

So you could really start out with a bad beat here /images/graemlins/smile.gif

Enjoy

MidGe

"Our human race is affected with a chronic underestimation of the possibility of the future straying from the course initially envisioned" - Nassim Nicholas Taleb

09-08-2005, 09:45 AM
[ QUOTE ]

Yes, or 10000, or 100000... etc.

It gets worse: there is as much chance to get such a sequence, beginning from your first observation, as any other specific sequence you may choose.

So you could really start out with a bad beat here /images/graemlins/smile.gif

[/ QUOTE ]

Yes its a perculiar how probability works. A BJ card counter might find the ultimate table that gives him an incredible 5% edge against the house. Yet he run the risk of being wiped out by a long chain of losers right from the start! If he restarts with a fresh pot after that, he still may get wipe out right from the first hand!

Gambling is indeed gambling. /images/graemlins/shocked.gif

For a martingale player on Roulette even chances, his chances of getting 10 losers in a row is 100% - 99.83% or 0.17%. A 0.17% risk of ruin.

The question is...BJ card counters run about a 1% risk of ruin (just guessing, not sure kekeke) even with a positive edge. Does this mean a card counter is as likely to go broke as a martingale user? Is Professional Gambling a myth? With more than 0.17% risk of ruin we're better off doing martingale (which doesn't work by the way)?? /images/graemlins/confused.gif

09-08-2005, 11:53 AM
And if a million people (who are immortal) repeatedly shuffle a deck of cards simultanously, eventually there will be a moment where all 1,000,000 of them by co-incidence manage to produce the same order of cards at the same time.

Infinity is not a real concept however - and I don't buy lottery tickets, either

09-08-2005, 11:59 AM
another thing

Suppose you are playing roulette and the numbers come up in this order:

4,12,8,34,12,30,19,5,20,15,7,22,9,3.

You wouldn't think this was anything out of the ordinary, right? It just looks like a typical random selection, doesn't it?

But now calculate how likely it is that this exact sequence shows up ...!!

/images/graemlins/cool.gif

LetYouDown
09-08-2005, 12:04 PM
[ QUOTE ]
But now calculate how likely it is that this exact sequence shows up ...!!

[/ QUOTE ]
1/13090925539866773438464

09-08-2005, 12:31 PM
Point being that the questioner appears to believe that some very long sequence of all red numbers is much less likely than some equally long but specific sequence that contains a mixture of reds and blacks.

OrangeKing
09-08-2005, 02:59 PM
[ QUOTE ]

For a martingale player on Roulette even chances, his chances of getting 10 losers in a row is 100% - 99.83% or 0.17%. A 0.17% risk of ruin.

[/ QUOTE ]

You've way oversimplified risk of ruin. You only have a 0.17% ROR on any 10 spins in a row...but presumably, your goal isn't just to win one unit or go bust. However, when you keep doing this over and over again, you only need to hit that rare event once to go bust..and that "rare" event becomes more and more likely to happen somewhere along the line the more spins you are going to play.

SheetWise
09-08-2005, 03:21 PM
I don't believe that anything that can happen will happen. In all games you can define a sequence of events whose probability is so close to zero, the probability can be substituted with zero.

On a regulation PGA golf course, it is possible that a player can shoot an 18. I put that probability at zero. I will offer any odds you want, and include every game played in the world for the next year.

LetYouDown
09-08-2005, 03:30 PM
[ QUOTE ]
On a regulation PGA golf course, it is possible that a player can shoot an 18. I put that probability at zero. I will offer any odds you want, and include every game played in the world for the next year.

[/ QUOTE ]
For a year? No. That wasn't the question. Add in the fact that we were referring to things that have a predetermined probability &gt; 0, and there you have it.

Infinity is like...a totally big number.

SheetWise
09-08-2005, 04:47 PM
[ QUOTE ]
Add in the fact that we were referring to things that have a predetermined probability &gt; 0, and there you have it.

[/ QUOTE ]
I don't understand. A player taking a shot at a golf hole has a probability &gt; 0 of hitting the shot into the cup.

If you want another example that can be calculated -- I think the probability of you picking the six numbers (single pick) that win the Powerball Lottery in three consecutive draws is zero. (Actual 1:[147,107,962^3])

Siegmund
09-08-2005, 05:56 PM
Our minds tend not to be well-equipped for the astronomical and the infinitesimal. Unusual things happen when both of these factors come into play.

Returning to the original question: yes, given a sufficiently large number of trials, anything that can happen with positive probability on one trial will eventually happen. On the other hand, given any fixed number of trials, you can find events so fantastically rare that it is almost impossible for them to occur within that time frame.

To take your golf example - at least on a short enough course that reaching the green in one shot is physically possible on every hole, there is some positive but small probability of a series of aces happening. (Say 1 in 1000 per hole, for the sake of argument - published estimates for pros on short holes range from 1500:1 to 4000:1. Less for amateurs or on long holes of course.)

Two consecutive holes in one is literally a one in a million (or one in a few million) chance, something you expect to make the news when it happens, but something you can expect to happen somewhere in the world reasonably often. A quick google search reveals it has indeed happened several times. There is argument about whether three in a row has ever happened. That's getting down to once in a lifetime or once in several lifetimes chances, but still not 'impossible'.

In real-life terms, bucking 1 in 10^54 or more odds to hit 18 in a row is getting up to the range where it takes an entire universe full of golfers playing for eons before it will ever happen. As long as you are talking about golf, you can call that "impossible" safely.

On the other hand, imagine you are interested in collisions between air molecules - interested in two particular types of collisions at very specific angles and speeds, so that collisions meeting your requirements happen exactly as often as winning Powerball 3 times in a row (1 in 3x10^24) or as hitting 18 holes in one (call it 1 in 10^54 for this example - but maybe really as bad as 1 in 10^70.)

At the pressures and temperatures at the surface of the earth, each molecule in the air bumps in to another one about 2 billion times a second. In just the bottom-most kilometer of the Earth's atmosphere, there are about 3.5x10^43 molecules bouncing around. The "as rare as eighteen consecutive holes in one" event, then, happens somewhere near the surface of the earth an average of once every couple minutes. The "3 in a row Powerball Winner" rare event happens about 10,000 times per second in every cubic centimeter of air, not so rare at all when compared with some of the things they build detectors for in particle physics.

I guess my point is that "rare" is a relative term. There are things you will see only a few times, things you are unlikely to ever see, things that noone is likely to ever see, things that are unlikely to have happened anywhere in the universe since the beginning of time.

LetYouDown
09-08-2005, 05:58 PM
[ QUOTE ]
I think the probability of you picking the six numbers (single pick) that win the Powerball Lottery in three consecutive draws is zero. (Actual 1:[147,107,962^3])

[/ QUOTE ]
But you're wrong. Infinity * (any number &gt; 0) = Infinity. The golf course question is debatable, but only in that there are physical limitations. It's impossible to hit a golf ball 600+ yards with current physical standards.

cwes
09-08-2005, 06:07 PM
The really funny thing is: If you spin a roulette wheel infinite times, there will be red streaks, black streaks and even green streaks of infinite length /images/graemlins/crazy.gif

SheetWise
09-08-2005, 07:10 PM
Excellent post. I'm really starting to hate these threads that contain the words "if" and "infinite".

LetYouDown
09-08-2005, 07:53 PM
[ QUOTE ]
Excellent post. I'm really starting to hate these threads that contain the words "if" and "infinite".

[/ QUOTE ]
Then don't debate them? LOL, in all likelihood, golf won't exist in its present form long enough for the stated example to even be a possibility. This entire thread revolves around infinity...if you're not a fan, there's no need to reply.

Dan Mezick
09-08-2005, 08:06 PM
If you gamble, you dont need to know this.

If you gamble WITH the odds, however, you need to accept the axiom that "anything that can happen will happen".

Traders call these events "shock events". For example an accounting scandal news release results in a 35% haircut on share value, gapping the stock lower on the open. That's a shock event.

The only way to protect from shocks is to limit your maximum bet size, such that Risk of Ruin (which cannot be 100% eliminated,) is acceptable to the trader.

Accepting the risk that anything can happen and playing accordingly is taking responsibility for all your results.

SheetWise
09-08-2005, 08:55 PM
[ QUOTE ]
I'm really starting to hate these threads that contain the words "if" and "infinite".

[/ QUOTE ]
[ QUOTE ]
Then don't debate them? LOL ... This entire thread revolves around infinity...if you're not a fan, there's no need to reply.

[/ QUOTE ]
Actually, LetYouDown, the OP did not mention infinity ... you introduced it. The OP mentioned a color on a roulette wheel occuring 100, 500, 1000 times -- that's a question that can be put into perspective using lifetimes. The constant need of some posters to introduce infinity into every problem is amusing. The concept of infinity totally ignores risk tolerance.

bobman0330
09-08-2005, 09:09 PM
Choosing .5 from the interval [0,1] can happen, but will not happen over an infinite number of trials. Correct?

LetYouDown
09-09-2005, 08:44 AM
How can you not factor in infinity when reading the initial post? That's assuming you've read it, however. To say that the probability of winning 3 powerball lotteries in a row is equal to 0 is ignorant. I assume you've never taken calculus or covered any course material dealing with limits?

SheetWise
09-09-2005, 10:25 AM
[ QUOTE ]
From OP:Is this correct? Since the probability cannot be reduced 0%, even with 0.0000276% chance of getting 20 reds in a row we will and do get 20 reds in a row. Does it mean that there is even a possibility of 100 reds in a row? Or even 1000 reds in a row?

[/ QUOTE ]
[ QUOTE ]
How can you not factor in infinity when reading the initial post? That's assuming you've read it, however. To say that the probability of winning 3 powerball lotteries in a row is equal to 0 is ignorant. I assume you've never taken calculus or covered any course material dealing with limits?

[/ QUOTE ]
When I read about the probability of unusual and unfortunate events in a "gamblers" forum, I assume it is related to risk tolerance. When you introduce "infinite" trials, you can show that any finite bankroll will come to ruin even in +EV games. This is then "corrected" by introducing the concept of the "infinite bankroll" ...

Siegmund hit the nail on the head.

You think that it makes sense to factor infinity into this problem of 1000 red spins in a series, in a game that at best can produce 100 trials per hour, in an effort to prove that it can and will happen. You then say that it is ignorant to disregard a 1:147,107,962^3 event in a game that provides two trials per week.

I can appreciate the comments in the thread that note these unusual events might happen in the first series of play, but so what? People take risks of greater magnitude all the time, flying, pressing a brake pedal, walking on a sidewalk, etc. Put in perspective, are you suggesting that what we consider to be acceptable risk with our life on a moment to moment basis would be an unacceptable risk with our fortune? Because "it can and will happen"?

Even casinos recognize how unusual the events are from the OP -- that's why they've set limits. They recognize that even their substantial, yet "finite" bankrolls may not survive the non-occurrence of these events.

LetYouDown
09-09-2005, 10:35 AM
We're not disagreeing on the likelihood. I interpreted the question literally, as "can and will happen". On a long enough time frame, it will happen. Is it likely? Absolutely not. Will it happen in my lifetime? Probably not. Siegmund, as usual, provided a real life example and interesting/intelligent explanations. It is unfortuante that he doesn't post more often. Probability is often theoretical, and I don't think that discussing the concept of infinity, which factors into every statistics/probability course, is unreasonable. I know where you're coming from with the risk tolerance assessment, but I didn't necessarily infer that from the OP.

SheetWise
09-09-2005, 11:04 AM
[ QUOTE ]
Is it likely? Absolutely not. Will it happen in my lifetime? Probably not.

[/ QUOTE ]
I suggest that the 2+2 forum coin a term representing a unit to use in describing unusual events. Many lay people use "... as likely as being hit by lightening" as a comparison -- which I find meaningless (I have no reference). Maybe if we used something like a unique nanosecond in an average lifespan or a unique minute in a century. We then call it Big, we can reduce large numbers to X number of Big's. Maybe it would make sense to use a number of game trials (B&amp;M) that a person could experience in a lifetime. I think lifetimes would be something everyone could relate to. Has this been done? Any ideas?

SheetWise
09-09-2005, 11:21 AM
[ QUOTE ]
I think lifetimes would be something everyone could relate to. Has this been done? Any ideas?

[/ QUOTE ]
I'm reponding to myself. As an example, for a coin flip to come up heads 100 times in a row, we would get 1:1,267,650,600,228,229,401,496,703,205,376. If we defined a roulette lifetime as 60 years of play, 12 hours a day at 100 trials per hour, a roulette lifetime would equal 26,280,000 trials. The 100 in a row sequence could then be reduced to 48,236,324,209,597,770,224,379 lifetimes. I think we would need a bigger number.

09-10-2005, 12:34 AM
Black Swan = term for highly unusual events in finance. coined by Prof Nassim Taleb who wrote the book "Fooled by Randomness".

Hmmmm....

1) According to some who replied, the probability of a black swan occurring can be so low that it doesn't happen in a lifetime.

EXAMPLE:
30 reds in a row in Roulette = 0.000000000176. The chance of this not happening is 99.99999998%. For a trial of 2000 spins, this will probably not happen. For a trail of 10,000,000 spins however, this might happen.

2)On the other hand, some who replied says it is as likely to happen!

EXAMPLE:
30 reds in a row happening = 0.000000000176 and 30 random mixture of reds/blacks happening in a row = 0.000000000176 too! Since the random mixture happens, why not 30 reds in a row when they have the same probability? This tendency to see long strings of identical outcome as "unique" is known as Leptokurtosis i think.

3)Finally I present a paradox which might probably be the result of my lack of skill in mathematics. /images/graemlins/tongue.gif

In a North America Roulette wheel the chance of hitting a Red is about 47.3%. This refers to 473 Reds out of 1000 spins IN THE LONG RUN. (100,000 spins?)

The chance of 100,000 Reds happening in 100,000 spins is a positive number still but very very very small. (0.473^100000) So this could happen.

But if 100,000 Reds occur in 100,000 spins...wouldn't the probability of Reds be 100% instead of 47.3%? The 47.3% dictates that 473 out of 1000 MUST be Red in the LONG RUN. Just like the ~5% house edge dictates that gamblers WILL lose ~5% of their bet in the long run.

Will the occurance of black swans not contradict probability theory?? /images/graemlins/confused.gif

SheetWise
09-10-2005, 12:55 AM
[ QUOTE ]
Will the occurance of black swans not contradict probability theory??

[/ QUOTE ] /images/graemlins/wink.gif

09-10-2005, 01:36 AM
Dear remiraz,

[ QUOTE ]
Will the occurance of black swans not contradict probability theory??

[/ QUOTE ]

You are distorting the work of Nassim Taleb /images/graemlins/smile.gif

Black swan events are a particularity of non-finite (or not completely known) sets. It doesn't apply to poker which is a game with, a large, but finite number of states (its set).

Unless you would consider being dealt AA 100 time in a row, a back swan event. I would not, it is a distinct possibility. In fact, it is AS possible as ANY arbitrary sequences of hole cards being.

irchans
09-10-2005, 09:00 AM
If there are 10,000,000 spins a day for 1 billion years, then we can expect 49 reds in a row once. Thus I think we can safely say that 60 reds in a row will never occur on this planet (on a fair wheel).

SheetWise
09-10-2005, 09:12 AM
It's a manageable risk. /images/graemlins/wink.gif

09-10-2005, 10:58 AM
[ QUOTE ]

3)Finally I present a paradox which might probably be the result of my lack of skill in mathematics. /images/graemlins/tongue.gif

In a North America Roulette wheel the chance of hitting a Red is about 47.3%. This refers to 473 Reds out of 1000 spins IN THE LONG RUN. (100,000 spins?)

The chance of 100,000 Reds happening in 100,000 spins is a positive number still but very very very small. (0.473^100000) So this could happen.

But if 100,000 Reds occur in 100,000 spins...wouldn't the probability of Reds be 100% instead of 47.3%? The 47.3% dictates that 473 out of 1000 MUST be Red in the LONG RUN. Just like the ~5% house edge dictates that gamblers WILL lose ~5% of their bet in the long run.

Will the occurance of black swans not contradict probability theory?? /images/graemlins/confused.gif

[/ QUOTE ]

"The 47.3% dictates that 473 out of 1000 MUST be Red in the LONG RUN.", this is only true if by "the long run" you mean an infinite number of spins. Given 100,000 spins, there is no "must" involved, it is only probable that about 473 out of 1000 will be red.

09-14-2005, 09:41 AM
[ QUOTE ]
If there are 10,000,000 spins a day for 1 billion years, then we can expect 49 reds in a row once. Thus I think we can safely say that 60 reds in a row will never occur on this planet (on a fair wheel).

[/ QUOTE ]

anyone here seen one of those before? the look on gamblers betting black and losing all the way must be very amusing. /images/graemlins/grin.gif

09-14-2005, 09:50 AM
[ QUOTE ]
Thus I think we can safely say that 60 reds in a row will never occur on this planet (on a fair wheel)

[/ QUOTE ]

umm! fooled by randomness!

[see Nassim Nicholas Taleb]

09-15-2005, 08:33 AM
[ QUOTE ]

umm! fooled by randomness!

[see Nassim Nicholas Taleb]

[/ QUOTE ]

60 reds in the row has the same probability as 60 red/black mixed in a row right?

LetYouDown
09-15-2005, 09:19 AM
[ QUOTE ]
The 47.3% dictates that 473 out of 1000 MUST be Red in the LONG RUN.

[/ QUOTE ]
No, just that it's extremely likely to work out this way. It is not absolute.

[ QUOTE ]
60 reds in the row has the same probability as 60 red/black mixed in a row right?

[/ QUOTE ]
Any specific color pattern of red/black will have the same odds as another.

09-15-2005, 09:34 AM
Probability tells you that 60 reds in a row is as likely as any other sequence you may posit.

LetYouDown
09-15-2005, 09:41 AM
[ QUOTE ]
Probability tells you that 60 reds in a row is as likely as any other sequence you may posit.

[/ QUOTE ]
I posit 60 greens in a row.

09-15-2005, 09:42 AM
nh /images/graemlins/smile.gif

magiluke
09-15-2005, 10:14 AM
[ QUOTE ]
Is it true in probability theory that "anything that can happen will happen"?

[/ QUOTE ]

No.

You flip a coin until it comes up heads. Heads has a probablilty of .5.
You probably will get heads fairly quickly, but there is nothing that
states that you absolutely will get that heads. There is the possibility that
it could happen in a year. There is the possibility that you will never see it.
Of course these are improbable examples, but they could happen.

It's a little bit easier to understand if you look at the lottery:

Lets say you have a 1/5000000 chance of winning the lottery (I don't know
what the actual odds are, especially since lotteries vary, so I'm arbitrating
it somewhere above the actual chances, I think). You can win the
lottery 5 times in a row, but the chances are (1/5000000)^5 (http://www.google.com/search?hl=en&amp;q=%281%2F5000000%29%5E5&amp;btnG=Google+S earch), and
that's a very small number. I don't see that every happening, even though
it has the possibility to.

Of course, if you throw that up against infinite
trials, it will happen. But I'm not doing it infinite times. You aren't
doing it infinite times. No one is doing it infinite times. Infinity is
just a concept, and for the purposes of reality, doesn't exist. Only the
arbitrarily large exists.

09-15-2005, 10:23 AM
[ QUOTE ]
Of course, if you throw that up against infinite
trials, it will happen. But I'm not doing it infinite times. You aren't
doing it infinite times. No one is doing it infinite times. Infinity is
just a concept, and for the purposes of reality, doesn't exist. Only the
arbitrarily large exists.

[/ QUOTE ]

Indeed, I am also really concerned with reality... I am just not sure about where it starts, or where I am on this infinite sequence. And to say that my experience will be neccessarily(!) unremarkable, is not taking the real for real /images/graemlins/smile.gif

magiluke
09-15-2005, 10:37 AM
[ QUOTE ]
Indeed, I am also really concerned with reality... I am just not sure about where it starts, or where I am on this infinite sequence. And to say that my experience will be neccessarily(!) unremarkable, is not taking the real for real /images/graemlins/smile.gif

[/ QUOTE ]

That was far too poetic for me to understand. Maybe I'm dumb.

Say not big words for me know you say.

09-15-2005, 11:02 AM
[ QUOTE ]
That was far too poetic for me to understand. Maybe I'm dumb.

[/ QUOTE ]

OK,

First, I am nearly certain you are not dumb.

Now, trying to strip the poetry and the big words. /images/graemlins/smile.gif

Infinity is not a concept. Reality is a concept, a sub-set of infinity.

All sequences are possible infinitely,, 60.. 61... 59.. 62.. 58.. 69.. 61.. etc.. and recurrences (happen again) of those.

Why should we not currently experience reality in that part of the possible sequence that is as above?

If we did, we would say that the likelyhood of 60 is very possible...? It is just as arbitrary as for it t it settles on 50% ... LOL

/images/graemlins/smile.gif

Enjoy

magiluke
09-15-2005, 11:40 AM
[ QUOTE ]
Infinity is not a concept. Reality is a concept, a sub-set of infinity.

[/ QUOTE ]

I like that.

Anyway, what I meant is that 'reality' is not infinite. Just because reality doesn't have a foreseeable end doesn't mean that it is infinite.

A person, in reality, doesn't have the ability to play the lottery an infinite number of times. Firstly, their life span is finite. There are a finite amount of lotteries running a finite amount of games (usually one per day, I believe). The lotteries will exist for a finite amount of time. The person playing has a finite bankroll.

This is the reality I was speaking of. And for the purposes of this reality, and there can not be infinite trials.

I was just giving an example that can't be carried out infinite times, and has a chance of occurring roughly equal to 0.

As for my next example, let's look at the numbers that exist between 0 and 1. There are an infinite amount. Just say that we wrote a program to map out all of these numbers. In reality, this computer would not be able to do it. It also has constraints that disallow it to complete it's task. Most obviously are the memory limitations. Even if we had infinite memory, the system would eventually break down; the connections burn out, e&amp;. Then there are the irrational numbers, that are themselves infinite. The computers would take forever trying to compute just one of these.

That is what I meant, although I do see your point.

09-17-2005, 04:07 AM
If we bet on Red in Roulette. The chances of us losing on a North America (double greens) is abt 0.5263.

The chance of 10 losers in a row is 0.5263^10 or 0.163%

0.163% is about 1 in 650. So in the long run about every 650 rounds of 10 will get a 10 losers in a row. 99.837% of the time u'll get a mixture of wins + losses.

The funny thing is...i wrote a Red/Black/Green simulator and 10 in a row seems to appear far more often! Like 1 in 20/30 rounds or something!! /images/graemlins/shocked.gif

SheetWise
09-17-2005, 04:24 AM
[ QUOTE ]
The funny thing is...i wrote a Red/Black/Green simulator and 10 in a row seems to appear far more often! Like 1 in 20/30 rounds or something!

[/ QUOTE ]
Arizona ran a numbers game for years before someone noticed that the number zero never appeared. Seems the programmer applied the modulus operator incorrectly to the RNG results. The entire program couldn't have been longer than 5 lines, and he managed to screw it up. /images/graemlins/wink.gif