Probability and die

Question

Say I have a die, it has 6 faces with 1-6 on each face. Every time I throw it one of the numbers shows up with equal probability. Now we play a game, you pay me a fee to enter the game. I throw the die if it end up on 1 you get 1 dollar, if it lands on 2, you get 2 dollars and so on. What is the fair value to charge to play?

So I know this is $3.50

But if I were to throw the die once and you get the option to either keep that roll or roll it one more time what is the fair value to charge then?

This is where I need help

Answer 1

Assuming: the second roll has no additional cost attached and if you roll twice you must use the second roll.

As you correctly point out, the expected value for one toss is $3.5$.

Thus, if you get a $4,5,6$ on the first time, you keep it...as the expectation is lower if you roll again. If, you get a $1,2,3$ on the first roll, you roll again and in that case your expectation is $3.5$.

Thus: $$\frac 16 \times \left(3\times 3.5 +4+5+6\right)=\boxed {4.25}$$