T Nation

Classic Probability Case...Fair or Unfair

So since Tiribulus was asking about my views on probability I find cases like this interesting and if others do not I apologize for eating bandwidth.

So once upon a time coke cost only five cents a bottle. The drinking kind! So a vending machine would dispense one bottle for a nickel. Then horrors the price went up to six cents a bottle.

To solve the problem the sole machine on an island was set such that each time you put your nickel in there was a 1 in 6 chance you got nothing. A sailor was there one day only. He put his first nickel in and got nothing. He put his second in and got his coke and left the island the next day bitter about the unfairness of coke machines.

Is the machine fair?

What would Von Mises say?

How about Keynes?

[quote]groo wrote:
Is the machine fair?
[/quote]

Need more data to determine this

[quote]groo wrote:
What would Von Mises say?
[/quote]

The second nickle and the coke shared the same birthday

[quote]groo wrote:
How about Keynes?
[/quote]

He would want to know the employment status of the sailor

Try this Groo. A thing is more probable that is closer to certain than is one that is further from certain. In other words, where a thing(could be any or type) occurs on the scale between utter impossibility and utter certainty defines how probable it is. Agree or disagree?

[quote]groo wrote:

So once upon a time coke cost only five cents a bottle. The drinking kind! So a vending machine would dispense one bottle for a nickel. Then horrors the price went up to six cents a bottle.

To solve the problem the sole machine on an island was set such that each time you put your nickel in there was a 1 in 6 chance you got nothing. A sailor was there one day only. He put his first nickel in and got nothing. He put his second in and got his coke and left the island the next day bitter about the unfairness of coke machines.

Is the machine fair?

[/quote]

No. Because the machine does not represent to the buyer that there is a chance he will get nothing. It is false advertising if the supplier represents to the consumer purchase conditions that the supplier then fails to meet. It would reasonable to suspect that a dispensing machine offering bererages at an advertised price, is going to dispense said beverage once the payment has been made.

However if the advertising on the machine represented the machine as a game of chance - much like the machines where you get three chances to grab candy with a robot arm - then that would different. It comes down to what a reasonable man would believe are the conditions being represented by the supplier. I think both Von Mises and Keynes would advise this sailor to call the number on the side of the machine and report the fact that he paid for a coke and didn’t receive it. I’m sure the company would be happy to send him a postal order refunding his money. 8*)

[quote]SexMachine wrote:

[quote]groo wrote:

So once upon a time coke cost only five cents a bottle. The drinking kind! So a vending machine would dispense one bottle for a nickel. Then horrors the price went up to six cents a bottle.

To solve the problem the sole machine on an island was set such that each time you put your nickel in there was a 1 in 6 chance you got nothing. A sailor was there one day only. He put his first nickel in and got nothing. He put his second in and got his coke and left the island the next day bitter about the unfairness of coke machines.

Is the machine fair?

[/quote]

No. Because the machine does not represent to the buyer that there is a chance he will get nothing. It is false advertising if the supplier represents to the consumer purchase conditions that the supplier then fails to meet. It would reasonable to suspect that a dispensing machine offering bererages at an advertised price, is going to dispense said beverage once the payment has been made.

However if the advertising on the machine represented the machine as a game of chance - much like the machines where you get three chances to grab candy with a robot arm - then that would different. It comes down to what a reasonable man would believe are the conditions being represented by the supplier. I think both Von Mises and Keynes would advise this sailor to call the number on the side of the machine and report the fact that he paid for a coke and didn’t receive it. I’m sure the company would be happy to send him a postal order refunding his money. 8*)[/quote]

Ok the sailor would know sorry if I didn’t clarify.

His position would tend to be something to the effect of well it wasn’t fair for me, but I suppose if I lived on the island and were there long term it might be ok. This would exemplify an eclectic view of probability as regards to expected value.

Von Mises was a frequency dogmatist as regards probability. So he would consider the price to be fair as on average the price paid was correct.

Keynes was a belief dogmatist. He is somewhat close to what you are saying. He would say the overall lottery system was fair. But that there are many types of unfairness and that forcing the one day visitor to gamble at the only game in town for a coke is where the unfairness arises.

So for me the expected value of the return is fair. However because the cost is relatively meaningless at a penny a position like a Von Mises would hold is ok. I think the frequency dogmatist view weakens as the relative costs and complexity grow to where people are forced to play the only game in town in a system where the true expected value isn’t easy to determine. This is why as well I think people make many ill considered choices…not so much irrational as not understanding the true expected value of a proposition.

Tiribulus your statement is too vague for me. Make it more specific please.

[quote]groo wrote:<<< Tiribulus your statement is too vague for me. Make it more specific please.[/quote]Specifics by definition miss the point. This is where you an I talk past each other and where Kamui and I do not. You’re asking me about floors in the building and I’m asking you about the bedrock that even the foundation of the building is built upon.

[quote]Tiribulus wrote:

[quote]groo wrote:<<< Tiribulus your statement is too vague for me. Make it more specific please.[/quote]Specifics by definition miss the point. This is where you an I talk past each other and where Kamui and I do not. You’re asking me about floors in the building and I’m asking you about the bedrock that even the foundation of the building is built upon.
[/quote]

Then I would say something like this:

Ok for belief probability

Personal degrees of belief can be expressed by betting rates.(There are several ways of getting to this idea most of them fairly verbose. I am not going to defend this or try to explain it if you disagree. You can say I am dogmatic on this one and choose to differ frequency dogmatists don’t even agree belief probability is real to some extent.)

Personal betting rates should be coherent.

A set of betting rates is coherent if and only if it satisfies the basic rules of probability.

Thus:
Personal degrees of belief should satisfy the basic rules of probability.

So we can express Bayes rule as this gobblygook for a partition of mutually exclusive and exhaustive hypotheses(H_i).
Pr(H_j/E)=(Pr(H_j ) Pr(E/H_j ))/Sigma[Pr(H_i ) (E/H_i )} with E as new evidence and H_i as competing hypotheses.
Pr(H_j) would be our initial probability before any new evidence.
E is new evidence
Pr(H_j/E) Would be our new probability in light of said evidence.

Which really is a kinda fancy way of saying we should learn from experience.

False advertising.

[quote]groo wrote:
So since Tiribulus was asking about my views on probability I find cases like this interesting and if others do not I apologize for eating bandwidth.

So once upon a time coke cost only five cents a bottle. The drinking kind! So a vending machine would dispense one bottle for a nickel. Then horrors the price went up to six cents a bottle.

To solve the problem the sole machine on an island was set such that each time you put your nickel in there was a 1 in 6 chance you got nothing. A sailor was there one day only. He put his first nickel in and got nothing. He put his second in and got his coke and left the island the next day bitter about the unfairness of coke machines.

Is the machine fair?
[/quote]
No, a coke costs 6 cents. So it either must give you coke for 6 cents every time or its ripping you off. For that machine, if set on a fix regiment of giving you a coke on the 6th try, then each coke costs 30 cents, not 6. So it’s a poor solution.

Keynes was an idiot, I don’t care what he says.

[quote]pat wrote:

[quote]groo wrote:
So since Tiribulus was asking about my views on probability I find cases like this interesting and if others do not I apologize for eating bandwidth.

So once upon a time coke cost only five cents a bottle. The drinking kind! So a vending machine would dispense one bottle for a nickel. Then horrors the price went up to six cents a bottle.

To solve the problem the sole machine on an island was set such that each time you put your nickel in there was a 1 in 6 chance you got nothing. A sailor was there one day only. He put his first nickel in and got nothing. He put his second in and got his coke and left the island the next day bitter about the unfairness of coke machines.

Is the machine fair?
[/quote]
No, a coke costs 6 cents. So it either must give you coke for 6 cents every time or its ripping you off. For that machine, if set on a fix regiment of giving you a coke on the 6th try, then each coke costs 30 cents, not 6. So it’s a poor solution.

Keynes was an idiot, I don’t care what he says.[/quote]

Orly?

Then read something besides his general theory on life, universe and everything and lo and behold, Keynes could write.

Anyone who writes the exact kind of drivel the market is waiting for is at least perceptive.

[quote]groo wrote:

[quote]Tiribulus wrote:

[quote]groo wrote:<<< Tiribulus your statement is too vague for me. Make it more specific please.[/quote]Specifics by definition miss the point. This is where you an I talk past each other and where Kamui and I do not. You’re asking me about floors in the building and I’m asking you about the bedrock that even the foundation of the building is built upon.
[/quote]Then I would say something like this:

Ok for belief probability

Personal degrees of belief can be expressed by betting rates.(There are several ways of getting to this idea most of them fairly verbose. I am not going to defend this or try to explain it if you disagree. You can say I am dogmatic on this one and choose to differ frequency dogmatists don’t even agree belief probability is real to some extent.)

Personal betting rates should be coherent.

A set of betting rates is coherent if and only if it satisfies the basic rules of probability.

Thus:
Personal degrees of belief should satisfy the basic rules of probability.

So we can express Bayes rule as this gobblygook for a partition of mutually exclusive and exhaustive hypotheses(H_i).
Pr(H_j/E)=(Pr(H_j ) Pr(E/H_j ))/Sigma[Pr(H_i ) (E/H_i )} with E as new evidence and H_i as competing hypotheses.
Pr(H_j) would be our initial probability before any new evidence.
E is new evidence
Pr(H_j/E) Would be our new probability in light of said evidence.

Which really is a kinda fancy way of saying we should learn from experience.
[/quote]Oh my merciful Lord Jesus help me with this young man =] If I find out you actually know where I’m goin with this and are intentionally stonewalling me? I will be very highly impressed. This is a simple question and yes there IS a simple answer.

Well, if you’re using this machine with the knowledge that you may walk away with nothing, then yes, it is fair.

I think both Keynes and Mises would say that it’s a stupid idea, as you’d drive away a lot of consumers this way.

[quote]pat wrote:

No, a coke costs 6 cents. So it either must give you coke for 6 cents every time or its ripping you off. For that machine, if set on a fix regiment of giving you a coke on the 6th try, then each coke costs 30 cents, not 6. So it’s a poor solution.
[/quote]

The machine fires a blank 1 out of every six times, not 5 out of every six.

You have an 83.3% chance of getting your coke for a nickel and a 16.6% chance of it costing you a dime.

Normally, 5 cokes would cost 25 cents. Now, they should cost 30. With this set up, they cost 25 cents plus a 5 cent blank, which evens out to 30.

If you know about the blank ahead of time, then it’s fair to the extent that it’s your choice and your risk. If it’s a secret, then it’s theft, which also could be fair depending on your definition of ‘fair’, I suppose.

[quote]TigerTime wrote:
You have an 83.3% chance of getting your coke for a nickel and a 16.6% chance of it costing you a dime. [/quote]
Nope.

Sorry - I’m just being an asshole, but there is a chance that it takes 3, 4 or more tries before you get anything. You assumed it could only take two or less. Moral of the story - probabilities are assholes, they are even more annoying than me

[quote]squating_bear wrote:

[quote]TigerTime wrote:
You have an 83.3% chance of getting your coke for a nickel and a 16.6% chance of it costing you a dime. [/quote]
Nope.

Sorry - I’m just being an asshole, but there is a chance that it takes 3, 4 or more tries before you get anything. You assumed it could only take two or less. Moral of the story - probabilities are assholes, they are even more annoying than me[/quote]

Sure but the main point of this is there are two main views on probability. Von Mises and other frequency dogmatists would find this pricing system completely fair, since in the long run the expected value would be correct.

Belief Dogmatists like Keynes didn’t particularly believe in a long run since in the end we all die and there is no long run for any of us.

Its the same as having a lottery where everyone buys in for five cents and 1/6 get nothing and 5/6 get 6 cents. It returns the proper expected value. The unfairness doesn’t derive from the lottery.

[quote]groo wrote:

[quote]squating_bear wrote:

[quote]TigerTime wrote:
You have an 83.3% chance of getting your coke for a nickel and a 16.6% chance of it costing you a dime. [/quote]
Nope.

Sorry - I’m just being an asshole, but there is a chance that it takes 3, 4 or more tries before you get anything. You assumed it could only take two or less. Moral of the story - probabilities are assholes, they are even more annoying than me[/quote]

Sure but the main point of this is there are two main views on probability. Von Mises and other frequency dogmatists would find this pricing system completely fair, since in the long run the expected value would be correct.

Belief Dogmatists like Keynes didn’t particularly believe in a long run since in the end we all die and there is no long run for any of us.

Its the same as having a lottery where everyone buys in for five cents and 1/6 get nothing and 5/6 get 6 cents. It returns the proper expected value. The unfairness doesn’t derive from the lottery.
[/quote]
I think this isn’t really anything about probability, but what one means by the term “fair”.

I would guess that Mises would say it’s fair as long as the buyer knows what the deal is before hand

I might guess that Keynes would say it’s unfair because some people pay more than others.

I’m not very familiar with Keynes though - so I’m just making that up. I also don’t remember reading about any Mises comments on this classic case either - so I’m making it all up.

But I’m pretty confident that what we are really dealing with here is semantics.

Groo is a very sharp kid, but has the attention span of a steamed turnip.

[quote]Tiribulus wrote:
Groo is a very sharp kid, but has the attention span of a steamed turnip.[/quote]

No I just mislike everything being turned into the same sort of argument.

I also dislike the rhetorical technique you seem to use unknowingly of asking someone to express something simply and coversationally like you simply want to know what they think and then trying to mine away at it like its some rigorous proof.

I don’t think we are all rational actors at all. I think there are many times where we do act irrationally knowing we do so. These times aren’t going to be changed by weighing each choice for expected value and return. Things like playing the mega millions when its huge for example. This is a horrible expected value and the game is rigged. Its totally irrational to play. I think the comparison was something like if you stacked pennies to the moon the chance of winning was the same as you grabbing one single penny out of said stack. I totally bought 20 dollars worth of tickets knowing this :P.
I still also believe that someday I will have a wild affair with Angelina Jolie another unlikely event…though probably more likely than winning the mega millions.

There are many other situations though where the stake matters a bit more that we have faulty information or aren’t processing the information we do have where we attempt to be rational actors and are failing because of that. In complex situations this is often why we make a poor choice.

Also there situations that are very complex to model like say the weather where a percentage chance is given that is often wrong. Also sometimes with weather and some other systems there is a tiny chance of something that when it does occur it is so devastating that likely it should have been taken into account. New Orleans for example.

So when you give a very vague question that you might try to attack a specific answer to can you see why it might be discounted?

Also I didn’t respond to the idea that these things weren’t questions of probability because I said I wouldn’t defend the idea they were expected value propositions.

[quote]groo wrote:<<< No I just mislike everything being turned into the same sort of argument >>>[/quote]Everything IS the same sort of argument. Until that argument is settled. I do wish there was a way I could say this without you taking it as sneering disdain because that is most definitely not the case. Sincerely. However, at this season of your life and study you appear simply unequipped to think in a truly foundational manner. EVERYTHING you want to discuss is immutably governed by and emerges from everything I want to discuss.

[quote]Tiribulus wrote:

[quote]groo wrote:<<< No I just mislike everything being turned into the same sort of argument >>>[/quote]Everything IS the same sort of argument. Until that argument is settled. I do wish there was a way I could say this without you taking it as sneering disdain because that is most definitely not the case. Sincerely. However, at this season of your life and study you appear simply unequipped to think in a truly foundational manner. EVERYTHING you want to discuss is immutably governed by and emerges from everything I want to discuss.
[/quote]

Your last two posts have been insulting in tone. Need I go back to posting troll gifs? That is all they deserve.

I think in a pragmatic manner in most things as I said before. Probability works for many things. Many things its unsuited for.

If you thing an imaginary voice guiding you is the best way to achieve expected value in your life choices so be it. I don’t need every debate bogged down in this. At our foundation the premises are too different.

In my opinion there is no need to defend probability and expected value in outcome from any claim that would begin its assertion well its only so from the divine grace of god.

There are plenty of grounds to not agree with belief probability or even that it exists, that is not one of them. Every topic doesn’t need to get to this.

You certainly may have some supernatural channel of knowledge its possible. I don’t believe it to be the case, but I have been wrong before…as have you.