# Expected Value and Fair Value clarification (work already done)

#### JohnWayne

##### New Member
4. In the lottery game FANTASY 5 you have to select 5 numbers from the numbers {1,2,3,... ,38,39} for a jackpot of $40,000. Each ticket costs$1. In this lottery each number must be
different and the order of the numbers does not matter.
a. What is the probability of winning the jackpot?
b. What is the expected value of this game?
c. How large does the jackpot have to be for it to be a fair game?

a. 1/575,757
b. E(X) = $40,000(1/575,757) + 0(575,756/575,757) = 0.06947 Now, my professor subtracted the$1 it cost to play the lottery game, so E(X) = 0.06947 - $1 = -$ 0.9305

c. This one is confusing to me. So the expected value of my $1 bet is (0.06497 or -0.9305)? then what is the fair value? My professor said that the fair value is$0 because the expected value is $0. Why is it not$1?

AND how large does the jackpot have to be for it to be a fair game? He wrote down $575,757 isn't this not fair? because I put in$1 and expect a return that is less

Thanks!

#### JohnWayne

##### New Member
"A game is said to be 'fair' if the expected value for winnings is 0" can someone please give me the intuition behind this? I can't seem to understand it :/

#### asterisk

##### New Member
When the jackpot is $575,757 E(X) =$1 - $575,757(1/575,757) + 0(575,756/575,757) = 0 If you play the lottery 575,575 times, you would expect to win 1 time. You would win 575,757 and your tickets would cost$575,757, so you would break even.

E(X) = $575,757 -$575,757 * 575,757/575,757 = 0

The house does not have an advantage and you do not have an advantage. The lottery is fair.