Expected value of die rolling game.
I was told that this is how to solve the following problem, but I just can't totally buy it and get my head around it. So if any of you can verify that this is correct, I would be indebted to you.
You've got a fair die, and I propose the following game. If you roll a 1 or a 2, I will give you 10 dollars (and the game ends). If you roll a 3 or a 4, you get another roll but the prize is discounted by d. That is, if you roll four 3's in a row, then roll a one, I give you 10d^4. If you roll a 5 or a 6, the game is over and you get nothing.
I want to find the expected value of this game, lets call this expected value G. The way I was told to solve is the following. There is a 1/3 chance you a 10 dollar prize, a 1/3 chance you get another roll (discounted) and a 1/3 chance you get nothing. Hence....
solving for G we get...
Is this correct? Thank you kindly for any advice.