Solution to ENVELOPE GAMBLE I

here is the possible outcome in a tabular form
| x | 2x
--------------------------------
dont change | x | 2x
--------------------------------
change | 2x | x

so if i dont change my "profit" obviously is going to be 0.

but if i change my "profit" is going to be 100% in one case and -50% in the other case. that makes the average profit to be 25%.

I am a bit confused about the approach, as intuitively i feel as this is same as suggested by the question so might be the wrong approach.

what do you think,

Update: another solution suggested is why should we consider the gain %age(as intially the amount is not with us, we are given some amount and an option to change it, %age wise though there is a gain if we change). So from the above table as we see, there is equal probabilty of getting x or 2x. Hence in this case if we change or dont change does matter.

Another observation: the answer might change if we say, (a) one amount is 1.5 times the other, or (2) one amount is 3 times the other.