Question

12. Suppose you pay $3.00 to roll a fair die with the understanding that you will get back $5.00 for rolling a 1 or a 6, nothing otherwise. What is your expected value?
$3.00
$-3.00
$5.00
$-1.33

Answer 1

In decimal value the answer is -1.33

E = 5(2/6) + (-3)(4/6) = (10/6)-(12/6) = -2/6 = -1/3

E = -1/3 or about -0.33333.

This is an unfriendly game because you are anticipated to lose 0.33 on every move in the long run.

E = (1/6) ・ 0 + (1/6) ・ 0  +  (1/6) ・ 5 + (1/6) ・ 0 + (1/6) ・5 + (1/6) ・ 0 - 3

E = 5/3 - 3

E = -(4/3)

Playing the game costs $3.

You win $5 if you roll 3 or 5, therefore you gain $2 ("+2").

If you roll a 1, 2, 4, or 6, you will not win anything and will lose $3. ("-3").

The predicted value is then calculated as

(2/6) ・(2) + (4/6) ・(-3) = 2/3 - 2 = -(4/3)

For every three games performed, you should expect to lose $4 in total.