# Expectation of changing the gift choice [closed]

Suppose we are given two boxes, with one of gift valued $n$ dollars and the other one valued twice as much. We can pick a box, and after open it we have the choice of switching to another box. Shall we switch?

I was asked about this during some graduate student seminar. My reasoning is since two choices are equally likely, it is equally likely I will get a box half of the original value and a box twice the original value. So I should switch as the net expectation is positive. However, this implies that my original choice is somehow inferior to the choice after switch, which violates symmetry principle. Can someone help?

If you did not see it: You are calculating the expectation wrong. You said there are two events, the second box containing $2v$ or $v/2$. But this conditions on a particular $v$ being in the first box. You did not see what's in the first box, so this is inadmissible. The correct pair of events is that either the second box contains $2n$ and the first box contains $n$, or else the second box contains $n$ and the first contains $2n$, with probability $0.5$ for each event. (Of course you are therefore indifferent to switching.)
If you did see it: You are calculating the expectation wrong. Conditioned on opening the box and seeing $v$, there is a posterior probability that the other box contains $2v$ and a posterior probability that the other box contains $v/2$. These probabilities are determined via a Bayesian update from their prior probabilities, updating on the event that the first box contains $v$.
In general these probabilities will not be $0.5$, but we cannot compute them because you did not tell us a prior distribution on the values in the boxes. Without a prior, the posterior is not defined and you cannot calculate the expected value of switching.
For example, if you are told $n = 50$ in advance, i.e. guaranteed that one box contains $50$ and the other $100$, then of course you should switch if the first box contains $50$ but should not switch if the first box contains $100$.
If you are told that $n$ is drawn uniformly from $[0,N]$, then you should not switch if you see $v > N$ because you must already have the larger box. Etc.