Agame is played using one die. if the die is rolled and shows 1, the player wins $ 10. if the die shows any number other than 1, the player wins nothing. a. if there is a charge of $ 2 to play the game, what is the game's expected value?

For each roll you start with paying 2 dollars and you only with 10 dollars one out of 6 rolls (on average). 

So the cost for one play is 2 dollars and your win is 10/6. 
Value is -2+10/6=-1/3 dollars

So you lose 1/3 dollars on average with each game.