i possibly misunderstood your question, do you mean like this?
E(value) = E(payoff - cost) =
Which simplifies quickly to what you wrote
You are to pay $1 to play a game that consists of drawing one ticket at random from a box of numbered tickets. You win the amount (in dollars) of the number on the ticket you draw. The following two boxes of numbered tickets are available.
Tickets have 0, 1, and 2
Find the expected value and variance of your net gain per play.
But since the game cost a dollar , correct?
Is there a way to factor the cost into the sum with out saying the sum is 1 minus the cost?