A lottery with 500 tickets gives one prize of 100 three priz

A lottery with 500 tickets gives one prize of $100, three prizes of $50 each, and five prizes of $25 each.

(a) Find the expected winnings of a ticket.

(b) If a ticket costs $1, what is the expected value of the game?

Solution

Expected Winnings = (P1st prize x X1st prize) + (P2nd x X2nd) + (P3rd x X3rd) + (Plose x Xlose)

P(winning 1st) = 1/1000 = 0.001 (there\'s only one $1000 prize out of the 1000 tickets)
P(winning 2nd) = 1/1000 = 0.001 (there\'s only one $500 prize out of the 1000 tickets)
P(winning 3rd) = 10/1000 = 0.01 (there\'s 10 out of 1000 tickets that will give you the $100 prize)
P(losing) = 988/1000 = 0.988 (if you don\'t win any of 12 possible prizes, then you\'ll end up losing, so 1000 - 12 = 988)

X(1st) = $1000
X(2nd) = $500
X(3rd) = $1000
X(lose) = -$3

Expected winnings = (0.001 x $1000) + (0.001 x $500) + (0.01 x $100) + (.988 x -$3)
= ($1) + ($0.50) + ($1) + (-$2.964)
= -$0.75
So you\'d expect to lose about 75 cents.

b) expected value of the game is 0.75-1 = -0.25