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Guest Message by DevFuse
 

plasmid

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Offline Last Active Yesterday, 05:55 AM
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Posts I've Made

In Topic: Whether to switch

10 May 2013 - 01:56 PM

I had been considering the scenario where the amount of money in the envelopes could be any real number. In that case, if you have no information about the probability distributions, both an integration of your post 13 over the entire range of possible values in the envelope and the experiment of making random probability distributions shows that there is no gain for switching.

However, I'm still not sure I can make an adequate math - to - english translation of those results; in particular showing how this is fundamentally different from a game where you are given $1000 and asked whether you want to flip a coin to either double or half your winnings (which is a no-brainer) in a way that makes intuitive sense.

If the amount of money in the envelope is restricted to integers, that's a whole new can of worms because being even or odd gives information. I'll have to mull over post 7 again and decide which of those two conclusions I like the best.

In Topic: Whether to switch

10 May 2013 - 04:09 AM

It doesn't matter what random number generation method you use, the results will be the same. And there's no need to code the program if you can tell by looking at it what the results would be.

In Topic: Whether to switch

09 May 2013 - 11:23 PM

The experiment is fairly simple. Randomly generate howevermany "smaller" envelopes you want {s1, s2, s3 ... sn} and for each of them generate a matching "greater" envelope {g1, g2, g3 ... gn} where the value in gx = 2sx. Let the value of all sx and gx be small relative to the bank account. The participant is given a random envelope and given the choice of whether or not to switch -- if he was originally given an envelope sx then he will switch to envelope gx, and if he was originally given an envelope gy then he will switch to envelope sy. Now compare what happens if the participant takes a strategy of switching vs a strategy of staying with the initial envelope.

 

The probability distribution is random, and results should generalize to any probability distribution that you face.

 

One could argue that it doesn't fit the OP because the player isn't presented a value of $1000. I would counter that the precise value that the player finds when he opens the envelope is arbitrary (you could multiply all of the s and g terms by any value you like) and would not affect the conclusions of the experiment.


In Topic: Whether to switch

09 May 2013 - 02:42 PM

I believe there is a good reason why a probability distribution was not stated in the problem.

I can think of an "experiment" that I believe would be considered satisfactory by most people, that doesn't depend on the probability distribution of how much money is in the smaller or larger envelope. The answer would be appropriate if our only information were that $1000 is small compared to the available bank account (ruling out the trivial possibility that the bank account is less than $3000 which would make it impossible for the other envelope to hold $2000 and obvious that you should not switch).

In Topic: Whether to switch

08 May 2013 - 03:02 PM

Morningstar: what would happen if you changed your program so the amount of money in the first envelope was random? And would that prove that it's always better to switch from whichever envelope you're looking at?

Bushindo: It's certainly better to not switch if you know that there is a ceiling for how much money could be in an envelope and you see an envelope containing more than half of that. But the problem doesn't make any mention of a ceiling. And it may very well be that you don't know how much money is in the bank account, or even if you did then you know that the amount of money in the envelopes is small compared to the size of the bank account but not precisely how small.

That might lead one down the road of looking for a probability distribution with an interesting property. But a complete answer to the question should take into account that the probability distribution could be anything.

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