Being Smarter Than A Fifth Grader

[ToadMan]

X is the out come of a random event... in this case, the case.

 

so p(X) is the probability that you will draw that value... in the initial setup the p(X) is 3.8% for any number, as the game goes on, since we are removing number without replacement that increases.

 

So for example when the game starts you have p(X) = 3.8% for all numbers and an E[X] = $131477.54.

 

If you open the first 6 and they are the 6 lowest (this in itself is statistically unlikely, but ok for my example), then p(X) for any number is now 5% and E[X] is $170916.25. If you get offered less, you should continue.

 

Next you would open 5, say the 5 highest: p(X) = 6.67% E[X] = 31221.67

Four: 4 lowest: p(X) = 8.3% E[X] = 38995.83

Three: 3 highest: p(X) = 11% E[X] = 10327.78

Two: 2 lowest: p(X) = 14% E[X] = 13178.57

One: 1 highest: p(X) = 16.6% E[X] =7041.67

And this continues.

 

Since the draws are independent, there probabilities are uniform, and the mean is the E[X].

 

The only flaw here is that you might not walk away with maximum money, but you would stop when the stats say you are being offered more than you should expect to make. I suppose the strategy could be refined to consider best case next round also. That would be you quit since the gain if you select only the lowest values does not push the E[X] about the offer.

 

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Turns out that some of the analysis here isn't as good of a back of the envelope as I had hoped. Check out http://tenser.typepad.com/tenser_said_the_...tatistics_.html

 

In extension to that I would give you one more tidbit, at the end of the game, when they have two cases and they ask if you want to switch you should:

 

The p(X) that that case was the high number when you picked it was 0.03, but now you have shown all the other cases to not be that high number. That means that you don't calculate it as 0.5 that either case has the number, you need to find P(X|all the failures) which will show that the P(X) for the case you didn't pick is much higher. It's called the Monty Hall problem http://en.wikipedia.org/wiki/Monty_Hall_problem

 

FYI my command of stats isn't as gripping as I would like, so I may be talking out of my ass.

 

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Ok, I'm wrong it's not the Monty Hall problem, since the origin of the second case is random, all the other cases where opened randomly.

Edited June 17, 2007 by ToadMan

[ecnarf]

This just turned into the best thread ever.

[bishopx]

Holy shit, I think Adam has just witnessed the second coming./

[ecnarf]

No. Just awesomeness.

[Ravenous megin-sane]

i hate that word

[Sparq]

Awesomeosity?

[ToadMan]

Sweet I'm causing people to make up words.

 

If you are really interested in that kind of stuff, it's all from an introductory stats class I am taking (well it's introductory honours but whatever)... there is probably way more fascinating things in upper levels of the subject... The company I work for also deals heavily in stats.

[ecnarf]

It's some pretty neato stuff but the only stats I might ever be taking would have to do with p-values and experimental uncertainty.

[ToadMan]

mmmm. fun stuff.

[Prometheon]

Yeah, Stats 201 was an amazing class. You learn so much cool (and applicable) stuff.

[ecnarf]

I'm watching this show again. Some bitch came on, won $5000 dollars because her "classmates" got three answers correct, and then didn't know that Alaska was larger than Texas.

 

Jesus fucking christ. Americans' ignorance of the world around them is only outdone by their ignorance of themselves.

[sodamntired]

Texas is pretty big. and Alaska is part of America.

 

you want to try that again?

[ecnarf]

Texas is pretty big. and Alaska is part of America.

 

you want to try that again?

My point has been proven.

[sodamntired]

hey, i know the internet allows some ambiguity, but if you keep going around posting your opinion about Hitler and the Holocaust like that, somebody is going to take offense.

[ecnarf]

Okay, let me put this for you quite simply:

 

Alaska > Texas

 

That's all there is to it.

Edited July 16, 2007 by ecnarf

[Azalroth]

Texas is pretty big. and Alaska is part of America.

 

you want to try that again?

Not sure where you going with this but I'm siding with Adam on this one.

 

Americans' ignorance of the world around them is only outdone by their ignorance of themselves.

 

This proves that he was talking about American ignorance in general...of themselves (aka Alaska AND Texas) and of the world.

 

Alaska is part of America.

Good call...not sure why you said it tho