I'd like to do multiple trials of the 2 envelopes problem so I can see the probabilities play out over multiple repetitions. I've never seen a description of how this happens and how the multiple trials relate to the logic in Wikipedia article that produces the paradox.
In the first round I put \$20 and \$10 in the 2 envelopes. My friend chooses the \$20 envelope and trades. We note the outcome.
Now we'll do trial #2. Do I use the same amounts as in trial 1 or can I use different amounts?
Wikipedia uses "A" in it's logic to denote the amount chosen. Is A a constant over multiple rounds? If so and the amounts change in each round, do I ignore the trials that don't include that amount A chosen in the first round and only use the ones in which that specific amount A is chosen so that I can see how the probabilities play out? Remember, I'm trying to see how the probabilities actually play out over multiple rounds. If I only use rounds that include the amount A, then it seems like I'll for sure do better trading because, indeed, about half the time I think the other envelope will contain 2A and the other half 0.5A and, maybe more importantly, I think this will show up when I do the experiment.
If A isn't constant, shouldn't this be addressed in the Wikipedia description of the logic used to determine the probabilities?
But the probability of putting A in an envelope twice is zero, right? That's because the probability of choosing any number from an infinite set of numbers is zero. If that's the case, doesn't that matter to the logic? I thought probability was a sort of average over multiple trials. What happens when the problem only allows something to happen once? Are there other problems in which an event can only happen once? I don't know of any.
Lastly, I thought the expected value would be the probability of ending up with a certain amount times the probability that the amount would be available to be drawn. The value times the probability. But that makes the expected value in each round zero because the probability of any particular value being offered is zero (choosing from an infinite set as mentioned above). So, total expected value is also zero. But, of course, it can't be zero. I'll end up with something every time I do a trial. What's wrong with this logic?
You can use the same amounts if you want or use different ones. Either is fine as long as the chooser does not know what the amounts are. If you use fixed amounts, say $10/20$, you should be able to calculate the chance you will switch with $10$ and with $20$ and all your simulation is doing is randomly choosing one of two biased coins and flipping it. You should observe that the chance you switch $10$ is greater than the chance you switch $20$. That is the heart of the proof.