Let us say you have a flash light. At full charge, it can last for 2 hours (or $T$.) Right now, it charge is a random variable $C$ which has a uniform distribution (or distribution $D$.) You may turn it on for any amount of time $O \ge 0$. During this time, if $C \ge O$, then $C_n=C-O$. If $C<O$, $C_n=0$, and you will learn $C_n=0$. After choosing $O$, and observing whether or not it dies, you have two options:
- Cancel trip. This has utility $0$ (or $Q$.)
- Go into the cave. If $C_n \ge 1$ hour (or $R$), you utility is $10$ (or $A_+$). If not, your utility is $-10$ (or $A_-$).
What procedure would output an $O$, take as input whether or not $C_n=0$ is learned, and then output "Go info cave." or "Cancel trip.", such that the utility is maximized.
Note: The extra variables are for those who want to generalize this procedure.
I would think for the uniform distribution $O=0$, and then which option you take is irrelevant, but I do not have proof.