Something I can't stop thinking about...

[Ratamahatta]

There is something I can't stop thinking about.

First: how do we define "probability of an event"? We define it as a limit of a ratio of events we interested in to total number of outcomes of a trial when number of trials goes to infinity. So it doesn't make sense to talk about probability of events when we only consider one single trial, right? If each posible outcome has a numerical value associated with it we can talk about EV and as gamblers we are interested in making bets with +EV.

Now consider this situation: someone get asked if he wants to make a single bet which is a +EV and he knows that it will be his only bet, although he will never make another gambling bet in his entire life. Should he take that bet? Intuitively you will say "yes". If the bet is +EV than it should be taken. But how can we say that the bet is +EV if it will be the only bet he will ever take? I say that the bet should not be taken because the concept of probability and EV does not apply here.

Am I crazy?

[jay_shark]

Just a little .

Of course the bet should be taken because it's positive EV .You may not live to enjoy the winning side of the bet but that shouldn't deter you from taking it . Any positive EV bet regardless of how often you take the offer is better than not taking it .

The number of hours we will play poker is finite .
The number of times we will brush our teeth is finite .
The number of times we will play the lottery is finite .
The number of hours we will breathe on earth is finite .

Are you implying that we should never bet on Tiger Woods to win another major because he's going to retire at some point ?

[binions]

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First: how do we define "probability of an event"? We define it as a limit of a ratio of events we interested in to total number of outcomes of a trial when number of trials goes to infinity.

[/ QUOTE ]

You are confusing probability with the law of large numbers, which says as trials approach infinity, results approach expected probability.

Mathematically, we know that 9 outs is 9/46 or 19.56% to hit with the river card to come (9/44 or 20.45% if two players are all in and both hands are turned over).

As trials approach infinity, results closely approximate the expected percentage. This is the law of large numbers.

[ QUOTE ]
So it doesn't make sense to talk about probability of events when we only consider one single trial, right?

[/ QUOTE ]

Wrong. We can know the probability without conducting infinity trials.

[ QUOTE ]
Someone get asked if he wants to make a single bet which is a +EV and he knows that it will be his only bet, although he will never make another gambling bet in his entire life. Should he take that bet? Intuitively you will say "yes". If the bet is +EV than it should be taken. But how can we say that the bet is +EV if it will be the only bet he will ever take? I say that the bet should not be taken because the concept of probability and EV does not apply here. Am I crazy?

[/ QUOTE ]

You so crazy.

If you flipped 1 coin 1 time, and I offered you $2000 if it was heads but you only had to pay me $1 if it was tails, would you take the one-time bet?

You are getting 2000:1 on a 1:1 (50%) shot. Does it matter that we only flip 1 time? No. We know the probability mathematically without having to run it infinity times.

[Ratamahatta]

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First: how do we define "probability of an event"? We define it as a limit of a ratio of events we interested in to total number of outcomes of a trial when number of trials goes to infinity.

[/ QUOTE ]

You are confusing probability with the law of large numbers, which says as trials approach infinity, results approach expected probability.

[/ QUOTE ]

I don't think I do. Probability of being delt aces in HE is 1/221, it means that after dealing large/infinit number of hands 1/221 of them will contain aces, nothing else. That's why it doesn't make any sense to talk about probability of a single event without thinking about infinit (or large/very large) number of trials. The same thing with EV. When we say that EV of a single bet is let's say $.5 when we bet $1, we are talking about what we expect to earn by making a single bet on average after making infinit (large) number of bets. Althoug the definition of probability and the law of large numbers are highly corelated.



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So it doesn't make sense to talk about probability of events when we only consider one single trial, right?

[/ QUOTE ]

Wrong. We can know the probability without conducting infinity trials.

[/ QUOTE ]


That's not what I ment. What I ment was that even if we know the probability of an event, theoretically it shouldn't make sense to use it for making a decision if we only consider one single trial because the deffinition of probability based on large/infinit numbers of trials. We still do it in everyday life, but theoreticaly speaking we shouldn't.


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Someone get asked if he wants to make a single bet which is a +EV and he knows that it will be his only bet, although he will never make another gambling bet in his entire life. Should he take that bet? Intuitively you will say "yes". If the bet is +EV than it should be taken. But how can we say that the bet is +EV if it will be the only bet he will ever take? I say that the bet should not be taken because the concept of probability and EV does not apply here. Am I crazy?

[/ QUOTE ]

You so crazy.

If you flipped 1 coin 1 time, and I offered you $2000 if it was heads but you only had to pay me $1 if it was tails, would you take the one-time bet?

You are getting 2000:1 on a 1:1 (50%) shot. Does it matter that we only flip 1 time? No. We know the probability mathematically without having to run it infinity times.

[/ QUOTE ]

Of course I would intuitively take that bet. What I mean is that since large/infinit number of trials is what definition of probability and EV are based on, it shouldn't make sense to use it while making a decision if we will never make another bet in our life.

meeeh, I am probably way off here...

[rufus]

What you're talking about is 'philosophical' in nature, and has been a subject of discussion for quite a while. The primary schools of thought regarding the interpretation of probability are, classical, Bayesian and frequentist.

The more classical notion of probability is based on finitely many equally likely cases, which works well for poker, but has mathematical limits.

The Bayesian model of probability is basically based on what you think will happen based on the information that you have.

The frequentist notion is based on the notion of repeatable experiments. In practice it's not necessary to repeat the experiment, but to be able to describe the 'experimental conditions'.

[ QUOTE ]
What I meant was that even if we know the probability of an event, theoretically it shouldn't make sense to use it for making a decision if we only consider one single trial because the definition of probability based on large/infinite numbers of trials. We still do it in everyday life, but theoretically speaking we shouldn't.

[/ QUOTE ]

There are mathematical definitions of probability, but they deal with set theory and analysis, and not with applications to the real world. In fact, the theory of probability was initially developed to allow people to make appropriate decisions in real life in gambling -- so your suggestion about not applying it is absurd.

Notably, the cited passage is self-contradictory - it assumes that the probability is known ("even if we know the probability"), and (2) that the probability cannot be known ("probability is based on a large number of trials").

"That word, I do not think it means what you think it means"
--Fezzig, The Princess Bride

It's unlikely to make direct practical difference in your life since you're apparently already willing to apply the theory of probability to decisions you make, but it might be rewarding for you to review your notions of probability. Self contradictory statements are often indicators of confused thinking.

[binions]

[ QUOTE ]

"That word, I do not think it means what you think it means"
--Fezzig, The Princess Bride


[/ QUOTE ]

INCONCEIVABLE!

IM - That Vizzini, he can *fuss*.
F - Fuss, fuss... I think he like to scream at *us*.
IM - Probably he means no *harm*.
F - He's really very short on *charm*.
IM - You have a great gift for rhyme.
F - Yes, yes, some of the time.
V - Enough of that.
IM - Fezzik, are there rocks ahead?
F - If there are, we all be dead.
V- No more rhymes now, I mean it.
F - Anybody want a peanut?

[binions]

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So it doesn't make sense to talk about probability of events when we only consider one single trial, right?

[/ QUOTE ]

Wrong. We can know the probability without conducting infinity trials.

[/ QUOTE ]

That's not what I ment. What I ment was that even if we know the probability of an event, theoretically it shouldn't make sense to use it for making a decision if we only consider one single trial because the deffinition of probability based on large/infinit numbers of trials. We still do it in everyday life, but theoreticaly speaking we shouldn't.

[/ QUOTE ]

Are you saying that even if we know the probability of an event without conducting a large number of trials, we can only use that probability knowledge if we envision multiple trials but we cannot use it if we only plan a single trial? That by conducting only a single trial, the chance of anything happening somehow magically becomes 50-50 (ie either it does or it doesn't)?

I don't see the hang up. You have over 2000 posts on a poker site. Single trial results don't matter. In the long run, repeated +EV decisions get the money they are supposed to get. And if that is true, it always makes sense to make a +EV decision, even if you will only make 1 wager for the rest of your life.

[tarheeljks]

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So it doesn't make sense to talk about probability of events when we only consider one single trial, right?

[/ QUOTE ]

Wrong. We can know the probability without conducting infinity trials.

[/ QUOTE ]

That's not what I ment. What I ment was that even if we know the probability of an event, theoretically it shouldn't make sense to use it for making a decision if we only consider one single trial because the deffinition of probability based on large/infinit numbers of trials. We still do it in everyday life, but theoreticaly speaking we shouldn't.

[/ QUOTE ]

Are you saying that even if we know the probability of an event without conducting a large number of trials, we can only use that probability knowledge if we envision multiple trials but we cannot use it if we only plan a single trial? That by conducting only a single trial, the chance of anything happening somehow magically becomes 50-50 (ie either it does or it doesn't)?

I don't see the hang up. You have over 2000 posts on a poker site. Single trial results don't matter. In the long run, repeated +EV decisions get the money they are supposed to get. And if that is true, it always makes sense to make a +EV decision, even if you will only make 1 wager for the rest of your life.

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they must be in bbv [img]/images/graemlins/laugh.gif[/img]. seriously though this is a little absurd.

[binions]

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they must be in bbv [img]/images/graemlins/laugh.gif[/img]. seriously though this is a little absurd.

[/ QUOTE ]

Not only should we make the single trial +EV bet, but assuming we know the payout odds and our chance of winning, the Kelly Criterion tells us how much we should bet of our available funds to maximize our return.

(Odds * Win%) - Lose% / Odds.

In the 2000:1 coinflip example, (2000*50 - 50) 2000 = 49.975% of our available funds should be wagered at 2000:1 on a coinflip.

[AaronBrown]

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But how can we say that the bet is +EV if it will be the only bet he will ever take? I say that the bet should not be taken because the concept of probability and EV does not apply here.

[/ QUOTE ]

Probability is a mathematical model, it's not reality. There's no mathematical answer to the question of which bets you should take.

In practice, if there are going to be a large number of independent bets, without too much variance in any one bet, the case for EV becomes compelling. If I offer to flip a coin with you, you get $2 for each head and pay me $1 for each tail, and we're going to do it a million times, and you're certain the coin is fair and the flips are independent, you're virtually certain to get within a few thousand dollars of $500,000 profit. It's theoretically possible you would not like this gamble, but pretty hard to imagine.

If you accept the case for large numbers, you could argue that if one million flips are good, each individual flip has to be good. After all, if something is bad, a million repetitions should make it worse. That's not unassailable logic, but it's pretty convincing for this case. It's even more convincing if you consider that in the first case I have given you an expected $500,000, which increases your wealth, which might make you more inclined to take gambles. So imagine that I give you $500,000 (you will need a good imagination, since I won't give you a quarter) and then ask if you'll flip a coin, heads you win $2, tails you pay $1. I think it's a pretty good case that you should say "yes."

Therefore I would say that if the person would take the bet if it were offered a large number of times, and we give him the expected value of a large number of repititions and then ask if he wants to gamble once, it's hard (but not impossible) to argue that he should turn down the gamble.