Question

A game is defined by the rules that two dice are rolled and the 
player wins varying amounts depending on the sum of the two dice rolled based 
on the following table:
Sum 2 3 4 5 6 7 8 9 10 11 12
Winnings $10 $9 $8 $7 $6 $5 $6 $7 $8 $9 $10
If it costs $7.50 to play this game:
a) what can a player expect to win by playing this game?
b)     Is this a fair game?if not, what would be a fair value to pay to play this game?   
Thanks

Answer 1

Prob(sum of 2) = 1/36
prob(sum of 3 ) = 2/36
prob(sum of 4) = 3/36
prob(sum of 5) = 4/36
prob(sum of 6) = 5/36
prob(sum of 7) = 6/36
... sum of 8 = 5/36
... sum of 9 = 4/36
... sum of 10 = 3/26
... sum of 11 = 2/36
... sum of 12 = 1/36

notice the sum of these prob's is 1

corresponding expections:
(1/36)(10) = .27777..
(2/36)(9) = .5
(3/36)(8) = .6666...
(4/36)(7) = .7777...
(5/36)(6) = .83333...
(6/36)(5) = .83333...
(5/36)(6) = .8333..
(4/36)(7) = .77777..
(3/36)(8) = .66666..
(2/36)(9) = .5
(1/36)(10) = .27777...

sum = $6.94

expected return = $6.94,
since it costs $7.50 this , like almost all gambling, is a losing adventure.