Question

3. A friend said that he was told that the expected value of a game was $1.00, but he played twice and lost $1.50. He said he must have been told a lie. How would you respond? Exercise 25 in Section 5.5 is relevant to this topic. What does it mean to say that a game is fair? Does it mean that you can never lose money? Exercise 18 in Section 5.5 is relevant to this topic.

Answer 1

The game is fair means there is no loss no profit or addition of loss and profit = 0 then that is fair game.

expected value = probability of winning * total profit after winning + Probability of losing * total profit after losing

So here expected value = 1 and he played twice ans lost 1.50 means in first chance he lost 0.75 $

Cost to play = 0.75

If there is half of probability is winning and half of probability is winning that 1/2

So 1/2 * (a-0.75) + 1/2 * -0.75 = 1

So a/2 -0.375 -0.375 =1

a/2 = 1 + 0.75 So a = 1.75*2 = 3.5

So winning amount = 3.5

Game is not fair because expected value = 1 not 0

So his friend is not correct, game owner is not lying.

winning prize is 3.5 but cost of play is 0.75 so expected value = 1 but if he lost two times he lost 1.50$