Problem - 4720
I can roll a die and collect the amount of money on the die, or if I don't like it, I can roll a second time and I have to pick up the die. What is my expected value?
The expected value of one-roll is $3.5$. So a rational player will only roll the second time if the first roll is below $3.5$. In another word, he will keep the first roll if it produce $4$, $5$, or $6$ whose expected value is $5$. Hence, the final answer is $$\frac{1}{2}\left(3.5+5)=\boxed{4.25}$$