I have a relatively simple problem for which I require an intelligent solution.
There are two rooms, each with 15 vases inside (30 in total).
Room A has a diamond hidden in one of the vases. This diamond is worth 1x
In Room B, there is also a diamond hidden in one of the vases, but this diamond is worth 2x
I'm allowed to search first in Room A and then in Room B, but I can only search inside 15 vases in total.
What is the optimal way to distribute my searches so that I'm always maximizing my profits? How can I extend this problem to diamonds with even bigger value differences? (i.e. 1x vs 3x)
Best, AC
This answer is wrong; I’ll be correcting it shortly.
“Maximizing my profits” isn’t a well-defined objective in a probabilistic setting. I’ll assume that you want to maximize the expected profit.
Every vase in room $A$ is worth an expected $\frac1{15}x$, whereas every vase in room $B$ is worth an expected $\frac2{15}x$, so you should search only in room $B$.
You might be confused by the prospect of finding the total value of $3x$ if you first search in room $A$, but by linearity of expectation you can’t gain from that in expectation. Whatever you gain in chances of finding the $1x$ diamond, you lose in chances of finding the $2x$ diamond, leading to a net loss in expectation.
This holds for any ratio of values of the diamonds.