Question

Suppose you pay $2.00 to roll a fair die with the understanding that you will get back $5.00 for rolling a 4 or a 2, nothing otherwise. What is the expected amount to win (or lose)

Answer 1

For a fair die roll, we have here:
P(2 or 4) = 2/6 = 1/3

Therefore, the expected payoff here is computed as:
E(Payoff) = 5*(1/3) = 5/3

Therefore the expected total profit / loss is computed here as:
= - Cost of fair die + E(Payoff)

= -2 + (5/3)

= -1/3

Therefore -1/3 is the required expected profit here while (5/3) is the expected payoff here.