>>13316297I think this question is not well formulated.
All probability questions assume an underlying probability distribution.
For example, "what is the probability to roll a an even number with a 6-sided die?" assumes a uniform discrete distribution from 1 to 6.
But here I'm not sure what underlying probability distribution should be assumed.
For simplicity, I'm going to assume that each envelope has to have a whole number of dollars.
A uniform distribution is often assumed when it's not specified. But there's no such thing as a uniform distribution across all natural numbers.
So we have two choices:
#1. We choose a uniform distribution from 0 to n for some n.
#2. We choose a non-uniform distribution across all natural numbers: p(n) = 2^(-n - 1) is such a distribution. (It's not the only one though).
Also, it doesn't matter for case #1, but for case #2 it matters if both envelopes are generated independently until one is the double of the other or if one envelope is generated and we then either put half or twice in the other with a 50% probability.
But once those questions are out of the way, we can stop doing English comprehension/philosophy and start doing math.
uniform distribution from 0 to n:
· If n < 10, then this scenario is impossible.
· If 10 <= n < 20, then you are guaranteed a $5 if you switch.
· If 20 <= n, switching gives you an expected value of $12.5.
· So you should switch if and only if you know that an envelope can have $20 or more.
2^(-n - 1) distribution, both envelopes generated independently until one is twice the other:
· The expected value of switching is around $5.00046, so you should not switch.
2^(-n - 1) distribution, one envelope generated and the other "made":
· The expected value of switching is around $5.44, so you should not switch.
But again, there are many other distributions that also agree with the wording of this question. You can probably find one where it won't matter if you switch or not.
