Something I can't stop thinking about...

[Ratamahatta]

Alright, I have difficulties expressing myself in english because I never studied the language and I can see that what I was trying to say was misunderstood. That's my own fault. I didn't mean that the bet shouldn't be taken, it was sort of an open question and I should have chosen a better example. I actualy study statistics and probability theory in university and I was trying to understand precise definition of probability and it's application because it's not as simple as you think it is. So let's live it for now, or just post it in bbv...

[img]/images/graemlins/heart.gif[/img]

[jay_shark]

Take the following example :

Let p be a successful outcome of an event, say flipping a bias coin with probability p=heads and q=tails . We flip this coin infinitely many times .

We need to check and verify that
p + qp + q^2p + q^3p +....+ = 1 . That is , the probability of hitting heads on the 1st trial , 2nd trial , 3rd trail , etc . The probability that this happens at some point should sum to 1 .

Factor out a p and we get a geometric series 1+q+q^2+...=1/(1-q) =1/p .
p*1/p=1 and so we're done .


Did I waste my time here ?

[rufus]

[ QUOTE ]
Alright, I have difficulties expressing myself in english because I never studied the language and I can see that what I was trying to say was misunderstood.

[/ QUOTE ]

I'm pretty sure that I understand the problem that you're having, which is something along the lines of: If probability says something will happen nine times out of ten (or something a bit more technical involving limits) how does it make sense to talk about a single trial that way?

A related question is:
If we shuffle a deck of cards, clearly, the top card we see is already determined, so how can we say the top card is each card of the deck with equal probability?

[AaronBrown]

[ QUOTE ]
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.

[jason1990]

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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 ]
In Euclidean geometry, the word "point" is undefined. Rather than being explicitly defined, it is characterized by the axioms of geometry. A point is any object which satisfies those axioms. Similarly, the probability of an event is simply a number between 0 and 1 which satisfies the axioms of probability. There is no mathematical definition beyond that.

What you are wondering about is the interpretation of probability. There are many competing philosophical interpretations of probability. The four major competitors are the frequency, propensity, subjective, and logical interpretations.

Here is a very loose description of the four interpretations. Imagine we flip a biased coin and we wonder about the probability of heads. According to the frequency interpretation, we can only talk about this probability if the flip is part of a long sequence of flips. In that case, the probability of heads is simply the limiting ratio of the number of heads to the number of flips. This is what you mentioned in your OP. If the coin is only flipped once and then subsequently destroyed, then according to the frequency interpretation, it does not make sense to talk about the probability of heads.

The propensity interpretation, however, does not require a long sequence of flips. Suppose we claim the probability of heads is 0.6. According to the propensity interpretation, this means the coin (or rather, the entire experimental setup, including the coin, the flipper, the air in the room, etc.) has a propensity for producing, in the long run, 6 heads for every 10 flips. It does not matter if the coin is destroyed after one flip. The probability of 0.6 refers to the potential frequency that would arise if we could flip it many times.

Both the frequency and propensity interpretations regard the probability of heads as describing some real physical property. With frequency, it is a property of the sequence. With propensity, it is a property of the experimental setup. The subjective and logical interpretations, on the other hand, do not regard probabilities as representing physical realities. Instead, they represent degrees of belief.

Suppose Joe says the probability of heads is 0.6. According to the subjective interpretation, this means that if Joe was offered a choice between these two bets:

(a) win $4 on heads, lose $6 on tails,
(b) lose $4 on heads, win $6 on tails,

then Joe would be indifferent as to which bet to take. The probability of 0.6 represents Joe's personal betting preferences regarding this event. Another person, say Jack, may have different preferences. Jack might, for instance, say the probability of heads is 0.3. In the subjective interpretation, neither is right or wrong. The statements they are making are not contradictory. They are simply subjective. This looks like it might be a totally useless interpretation, but there is one caveat which somewhat fixes this subjectivity. In the subjective interpretation, Joe's and Jack's probabilities must be "coherent," which means that they must obey the axioms of probability. These axioms ensure that if the coin is flipped many times, then Joe and Jack will no longer disagree about the probability of heads. Their subjective opinions will get closer and closer to each other, and in the limit of infinitely many flips, they will exactly agree on the probability of heads. In fact, their subjective opinions will match the frequency of heads to total coin flips. However, if the coin is flipped only once, then their opinions may differ dramatically, and (in the subjective interpretation) no one can say who is right or wrong, because in fact neither is right or wrong.

The logical interpretation is similar to the subjective. Probabilities represent degrees of belief, but they are not meant to be subjective. In the logical interpretation, probabilities arise because we have uncertainty about some proposition or event. This uncertainty exists because we have only partial information about the thing in question. In the logical interpretation, it is postulated that there exists some ideal form of reasoning that can be applied to this partial information which will yield a probability. In other words, if Joe and Jack have the same information about the coin flip, then they should arrive at the same degree of belief. In the logical interpretation, probabilities do not belong to the person stating the probability, but rather to the information which that person possesses.

Notice that it is only the frequency interpretation (the one you described in your OP) which forbids probabilities for one-time events.

Now, you also brought up EV and decisions. Probability theory (and science in general) cannot tell you what decisions you should make. It can only tell you the consequences of the various decisions you are considering. It is up to you to decide which consequences you prefer. Physics, for example, cannot tell us whether we should split an atom. It can only tell us what will happen if we do. The same is true of probability theory.

As far as EV is concerned, the relevant theorem is the Law of Large Numbers (LLN). The LLN, as you probably know, says that if you perform a sequence of independent and identically distributed wagers, then the overall average rate of change of your bankroll will converge to the EV of a single wager in that sequence. People often ignore the details of what the LLN says, and simply act as though it says you should always maximize your EV. But it is a mistake to ignore the details. In particular, it is mistake to ignore the fact that the LLN contains hypotheses which must be satisfied before it can be applied. In particular, the LLN requires not only that you have a sequence of wagers, but also that they are independent and identically distributed.

For example, suppose you are offered a 3:2 payout on the flip of a fair coin. What percentage of your bankroll should you wager on this bet? If you unthinkingly try to maximize your EV, then you will bet your entire bankroll. However, if you did this many times, you would eventually go broke. The LLN does not apply to a sequence of wagers in which you always bet a fixed proportion of your bankroll, because these wagers are not independent and identically distributed. The Kelly criterion will recommend a fraction smaller than 100% of your bankroll. If you accept Kelly's recommendation, then on a single wager, your raw EV will be smaller than it would be if you bet your whole roll. So Kelly, strictly speaking, is not recommending that you maximize your EV. If you study the Kelly criterion, you will find theorems that explicitly describe the long term consequences of following the Kelly system. The theorems do not say, "follow the Kelly system." They say, "if you follow the Kelly system, then here is what will happen." In some circumstances, the LLN is valid. In others, the Kelly theorems are valid. And in still others, neither might be valid and we may need to turn to something else in order to discover the consequences of our considered actions.

If you truly only make one wager ever, then the LLN simply does not apply. But in reality, you will probably make many wagers. If you define "wager" more broadly as any decision under uncertainty, then life is one long sequence of wagers. Even so, the LLN may not apply if the wager you are considering is not part of some subsequence which is independent and identically distributed. In that case, it may or may not be a good idea to maximize EV. The Kelly criterion illustrates a concrete example of this.

[Beermantm]

jason1990

First off thank you for that reply even though I'm not the OP I do a lot of reading in these forums but not a ton of posting. Matter of fact I would like to thank everyone that posted in this thread and for this whole topic. I consider myself a smart guy but I surely get humbled by the great minds on these boards. That answer was such an elegant well thought out and well written reply I had to post to tell you how much I enjoyed reading it.
Thanks
Phil