Question

One thousand chances are sold at $4 apiece for a raffle there is a grand prize of $650, two second prizes of $200, and five third prizes of $150. First calculate the expected value of the lottery. Determine whether the lottery is a fair game if the game is not fair, determine a price for playing the game that would make it fair.

a) $0.50
b) $1.00
c) $1.50
d) $2.00

Answer 1

The expected value of the raffle is -$2.20, indicating it is not a fair game. To make it fair, tickets would need to be priced at $1.80 each. The closest fair price option given is $2.00.

Step-by-step explanation:

To calculate the expected value of the lottery, we sum the products of the prizes and their probabilities, then subtract the ticket cost:

1. There's a 1/1000 chance to win the grand prize of $650.
2. There's a 2/1000 chance to win the second prize of $200.
3. There's a 5/1000 chance to win the third prize of $150.

So, using the formula for expected value:

EV = (1/1000 * $650) + (2/1000 * $200) + (5/1000 * $150) - $4

EV = ($0.65) + ($0.40) + ($0.75) - $4

EV = $1.80 - $4

EV = -$2.20

Since the expected value is negative, it's not a fair game. To make it fair, the price at which the expected value equals zero needs to be found:

0 = (1/1000 * $650) + (2/1000 * $200) + (5/1000 * $150) - X

0 = $0.65 + $0.40 + $0.75 - X

X = $1.80

Therefore, to make the game fair, tickets should be sold at $1.80 each, which is not an option provided. The closest answer that would make the game more fair is $2.00 per ticket.