Debra is buying prizes for a game at her school's fundraiser. The game has three levels of prizes, and she has already bought the second and third prizes. She wants the first prize to be nice enough to attract people to the game. The game's manufacturer has supplied her with the probabilities of winning first, second, and third prizes. Tickets cost $3 each, and she wants the school to profit an average of $1 per ticket. How much should she spend on each first prize? (Give your answer to the nearest cent.)

Question

Debra is buying prizes for a game at her school's fundraiser. The game has three levels of prizes, and she has already bought the second and third prizes. She wants the first prize to be nice enough to attract people to the game. The game's manufacturer has supplied her with the probabilities of winning first, second, and third prizes. Tickets cost $3 each, and she wants the school to profit an average of $1 per ticket. How much should she spend on each first prize? (Give your answer to the nearest cent.)

Answer 1

To determine how much Debra should spend on each first prize, we need to consider the cost of the tickets and the probabilities of winning each prize.

Let's assume that Debra sells 'x' number of tickets. Since each ticket costs $3, the total revenue from the ticket sales would be 3x dollars.

Now, Debra wants the school to profit an average of $1 per ticket. This means that the total cost of prizes should be the total revenue minus the average profit. So, the total cost of prizes would be 3x - x = 2x dollars.

Since Debra has already bought the second and third prizes, the cost of these prizes is given. Let's say the cost of the second prize is 'y' dollars and the cost of the third prize is 'z' dollars.

So, the remaining amount to be spent on the first prize would be the total cost of prizes minus the cost of the second and third prizes. Therefore, the cost of the first prize would be (2x - y - z) dollars.

Now, the probabilities of winning the first, second, and third prizes have been given. Let's say the probabilities are P1, P2, and P3 respectively.

To find the expected value of the ticket sales, we multiply the probability of winning each prize by its cost and sum them up. The expected value should equal the average profit, which is $1.

Therefore, the equation can be written as:
P1 * (2x - y - z) + P2 * y + P3 * z = 1

Given the probabilities, we can substitute their values into the equation to solve for 'x'. Once we find the value of 'x', we can calculate the cost of the first prize by substituting it into (2x - y - z).

Please provide the specific probabilities for winning the first, second, and third prizes given by the game's manufacturer, so we can proceed with the calculations.