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TomasB
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This paradox was inspired by Nir's post about (a+b)/(c+d) being between a/c and b/d and an old probability paradox. Thanks to Nir's explanation I found a way to pose the problem without posing it as a probability question.

--------------------------
I go to a store and find that they have by error marked two different prices on the apples: 67 cents and 64 cents.

Since I'm are an economical person I of course buy the cheaper kind. I buy 14 of them for 8.96 dollars.

The next day the prices are lower but again they have mismarked the same kind of apples. But now the prices are 45 cents and 43 cents. I do need some more apples, I realize, and again take advantage of the cheaper price and buy 7 of them for 3.01. So I've payed 11.97 for 21 apples for an average price of 57 cents an apple.

If we go through life always going with the best bargain, shouldn't that mean that we do better on the average than people who doesn't? And definitely better on an average than people that choose the lousiest deals?

I thought it did, until my friend said "I shopped apples there too, but I didn't see the lower price you spoke of."

First he had bought 9 apples for 67 cents a piece which costed him 6.03 dollars. The next day he bought 11 apples at 45 cents a piece for 4.95 dollars. Turns out that he had spent 10.98 dollars for 20 apples which is _less_ than 55 cents an apple and definitley less than the 57 cents per apple that _I_ had payed.

So he made a better deal on an average by going with the worst deals! What the heck is going on? Can this be true?

/Tomas
Steve Martin
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Yes, it is true. We think we are smart, but unless we change our ways we are all ultimately doomed. Smile
Any man who reads too much and uses his own brain too little falls into lazy habits of thinking.
Albert Einstein
magicjohn2278
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I know this one!

It's because today is Tuesday and today apples cost 99 cents each.

Your friend must go to the supermarket to buy his missing apple (you have 21 he has 20) - so now you each have 21 apples which cost you the same!

Am I warm?
TomasB
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Quote:
On 2006-03-14 08:43, magicjohn2278 wrote:
I know this one!

It's because today is Tuesday and today apples cost 99 cents each.

Your friend must go to the supermarket to buy his missing apple (you have 21 he has 20) - so now you each have 21 apples which cost you the same!

No, the total number of apples is not what is creating this "paradox". You can have both buy the same amount and still he comes out the average winner by using the bad deals.
Quote:
Am I warm?

Depends on how you are dressed and where you are.

/Tomas
magicjohn2278
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Tomas, I have exactly the same problem at my local supermarket... the solution to the problem is "Always count your change!" - Some of those checkout girls aren't too bright!

(14 a day? Don't you think that you might be eating too many apples?)
0pus
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I don't understand why this is a "paradox."
TomasB
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I think it's a "paradox" that you can do better deals by going with the worst deals than someone who goes with the better deals.

It's true that it isn't a paradox though since this is how it actually turns out.

/Tomas
TomasB
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I can explain it in a slightly different way with an example from real life.

At a party 5 out of a total of 11 girls with skirts are total sluts. Only 3 out of the 7 girls that wear pants are as willing. Since 5/11 > 3/7 I'm better off trying to pick up a girl with a skirt.

At another party 6 of the 9 skirted girls are sluts but only 9 of the 14 girls with pants, so again I'm better off trying to pick up a girl with a skirt as 6/9 > 9/14.

Later in the evening both these parties join at the same place, but that suddenly means that I'm better off trying to pick up a girl with pants!

I'd call that a "paradox", Opus, and we all should think in these terms from now on when we search love or bargains in apples. Smile

/Tomas
magicjohn2278
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Quote:
On 2006-03-15 15:32, TomasB wrote:
I'm better off trying to pick up a girl with a skirt.

At another .........., so again I'm better off trying to pick up a girl with a skirt as 6/9 > 9/14.

Later in the evening both these parties join at the same place, but that suddenly means that I'm better off trying to pick up a girl with pants!

/Tomas


Presumably your wife has no interest in computers!
TomasB
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She wears the pants in the family.

/Tomas
Daegs
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I still don't think this is a paradox....

The prices have to be looked at together rather than seperated based on days....If someone buys something at $1 and $2, then they have a $1.50 average. so someone could a buy something at $2.50 and a TON of things at $1.49 and still come up with the lower average because the lower one is weighted by the amount of things bought(well they are both weighted based on amount).

If they bought the same amount as eachother each day then it would really be a paradox... but as it stands this seems like a bad observation of simple math...(like the bellboy with the missing dollar).
TomasB
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It isn't a paradox. And using extreme examples like the one you mention is a good way to check it. It just gets a bit blurry when the amounts are about the same or, which you can do in this version, the total amounts are exactly the same. You can have them both buy a total of 20 apples and still the one going with the two worst deals will have gotten his 20 apples cheaper.

Maybe it's just my intuition that is faulty but

"If we go through life always going with the best bargain, shouldn't that mean that we do better on the average than people who doesn't? And definitely better on an average than people that choose the lousiest deals?"

sums up what my intuition told me.

Hmmm, maybe the probability version is better, come to think of it.

/Tomas
dlhoyt
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I couldn't find Nir's post, so I apologize if this is repetitious.

This counter-intuitive problem is a good example of Simpson's Paradox. A good discussion can be found at the following site:

http://plato.stanford.edu/entries/paradox-simpson/

This is no academic problem; there was once a sex-discrimination suit based on this type of calculation. Also, some approaches to the evolution of altruism vs. cheating make use of it.

Dale
TomasB
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Sorry, it was Landmark's post and explanation at

http://www.themagiccafe.com/forums/viewt......01&9

that inspired it.

/Tomas
Andrei
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Thomas - a socially intuitive 'explanation' for this paradox is, I think, as follows:

In life, it is wisest to make the most productive decisions possible with available resources. However, one of those decisions is actually choosing what decisions you want to be involved in!

In this particular case, it really doesn't matter if you spend 45 instead of 43 cents per apple, because both are a heck of a lot lower than 67.

So, in retrospect, the guy who has the most efficient buying pattern IS making the best decisions, because he made the correct decision of how many apples to buy in which days. That is actually the most important of all decisions.

I think, to philosophise a bit, this might be an important 'hidden' decision. The price differences can trick you into rushing and getting a better 'deal', when in fact you're worse off.

Andrei
TomasB
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Exactly, Andrei, which means that you are never sure you make the best decisions unless you plan your whole life ahead _and_ know the variables. Which of course is impossible. So we can find ourself doing worse deals than people that makes worse deals even though we go for the best deals.

/Tomas
Andrei
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Tomas (I realize I've been calling you Thomas, and apologize) - Well, it's obviously impossible to determine every variable, because uncertainty is a lovely part of life. And, obviously, luck plays a major part in anyone's success.

However, most of our choices are not selected by us. In other words, we are not the ones who make the decisions about which decisions we can make. The school you go to, for instance, is sometimes determined by your bank account, which is sometimes determined by your parents' wealth, which is not up to you.

Still, within the boundaries of these objective limits, we CAN have a winning strategy, because I like to believe that luck always evens out in the end. In other words, if you choose to pursue anything BUT the best short-term decisions, you will eventually end up worse in the long-run, because elements of luck such as those cited by your apple example eventually balance each other out.

That's what I like to think, anyway. If I were to think otherwise, it would be much more confusing, and not as pretty and enjoyable.

Andrei
Josh the Superfluous
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So where is this party?
What do you want in a site? "Honesty, integrity and decency." -Mike Doogan
"I hate it, I hate my ironic lovechild. I didn't even have anything to do with it" Josh #2
TomasB
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In Sweden of course. See you at the next FISM?

/Tomas
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