jimboston wrote:Well maybe $20 if I'm playing at a place that provides free drinks too...

Ah, they key to being a real winner in any gambling establishment. Make sure you cost them more in free booze than they take from you in gambling loses.

rdsrds2120 wrote:What a nice little paradox.

BMO

What a tease. I think there's more than one (pseudo)-paradox associated with this problem.

Elaborate plox.

x-raider wrote:Does the peek at the cash in the first envelope offer you any valuable info? Can this be used to increase your chances of getting the envelope with more money ? It depends. Are they different values? If so then I'd always swap from picking again from getting $1 at first. But either way, as the value of the first envelope goes up the chances of you receiving more than that in the 2nd diminish. At which point It would only be logical to settle for anything above $50. But I personally would settle for $35.

How about if instead of us having no information about the money in the envelope we knew for sure that the amounts in both were randomly chosen (from a uniform distribution, i.e. every value between $1 and $100 has a 1% chance of being in either envelope). Does this change anything? It changes everything. I assumed that already. I haven't considered the alternative. My brain will hurt if I try.

So the alternative would be that they come from ANY distribution.

Worst case scenario, this means the universe might be ruled by some malevolant deity that knows exactly what your strategy will be and what envelope you'll look at first and chooses sums to put in the envelope especially designed to f*ck up your plan. So, if we're not assuming the uniform distribution, worst case scenario, your plan would have to be "adversary-proof"(i.e. even if the deity existed and knew your plan you should still be able to come ahead somehow).

But it actually isn't too hard to extrapolate from the uniform distribution to the adversary distribution after you figure out the trick for the uniform (yeah, there's a trick).

Lootifer wrote:Not having encountered this before but it sounds very much like a simple form of blackjack.

Hard one to calculate; there are two possible reasons why the second envelope gives you a slight advantage.

Firstly the envelopes may not be independent; that is they may come from a cache of 100 envelopes (or some other finite number); thus by knowing one number you can infer estimates on the remaining population (this is the same principle on which counting cards in blackjack is based on).

Secondly the law of averages still applies in uniform distributions. The mean will still be 50.5 given enough samples; thus if your first envelope is less than this number the most rational decision is to select the second one. Yes you may lose even more by doing this (say you picked $47 in your first envelope and $12 in your second) but on average if you always pick based on this rule you will come out slightly better off.

Now I cant be bothered working out the impact of these effects, but the population is big enough for them to not mean much; i still wouldnt pay any more than $50.50.

Hey, you're back. I'm gonna have to find excuses to bump my previous math/statistics related threads now.

Your strategy is correct for the uniform case, but you're missing a big factor in how that strategy affects your estimated payoff. You should be willing to pay more.

Write a script if you don't believe me (I wrote a short one to convince myself ... wait do you write code? I forget)