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.)

Prize Cost of Prize Probability
1st $ 0.15

2nd $1.28 0.30

3rd $0.70 0.45

.15x + .3(1.28) + .45(.70) + 1 = 3

I got x = 8.67

Check:
expected value (what you have to pay out)
= .15(8.67) + .3(1.28) + .45(.7) = 1.995
or $2.00, but each ticket costs $3, so they make $1 per ticket

To find out how much Debra should spend on the first prize, we need to calculate the average profit made per ticket and then subtract the average profit from the ticket price.

First, let's calculate the average profit per ticket:
Profit per ticket = Ticket price - Cost of prize
Profit per ticket = $3 - Cost of prize

Next, we need to calculate the probabilities of winning each prize:

Probability of winning the 1st prize = 0.15
Probability of winning the 2nd prize = 0.30
Probability of winning the 3rd prize = 0.45

To calculate the average profit, we need to consider the probabilities of winning each prize and the profit per ticket derived from each prize:

Average profit = (Profit per ticket for 1st prize x Probability of winning 1st prize) + (Profit per ticket for 2nd prize x Probability of winning 2nd prize) + (Profit per ticket for 3rd prize x Probability of winning 3rd prize)

Since Debra wants the school to profit an average of $1 per ticket, we can set up the following equation:

$1 = (Profit per ticket for 1st prize x 0.15) + (Profit per ticket for 2nd prize x 0.30) + (Profit per ticket for 3rd prize x 0.45)

Now, substitute the expressions for the profit per ticket for each prize:

$1 = ($3 - Cost of 1st prize x 0.15) + ($3 - $1.28 x 0.30) + ($3 - $0.70 x 0.45)

Simplify the equation:

$1 = (3 - 0.15Cost of 1st prize) + (3 - 0.384) + (3 - 0.315)

$1 = 9 - 0.15Cost of 1st prize - 0.384 - 0.315

$1 = 8.301 - 0.15Cost of 1st prize

Rearrange the equation:

0.15Cost of 1st prize = 8.301 - $1

0.15Cost of 1st prize = 7.301

Cost of 1st prize = 7.301 / 0.15

Cost of 1st prize ≈ $48.67 (rounded to the nearest cent)

Therefore, Debra should spend approximately $48.67 on each first prize.

To calculate how much Debra should spend on each first prize, we need to consider the average profit per ticket.

First, let's calculate the average profit per ticket. The profit per ticket is given as $1. Since tickets cost $3 each, the average expenses per ticket can be calculated by subtracting the profit per ticket from the ticket price:

Average Expenses per Ticket = Ticket Price - Profit per Ticket
= $3 - $1
= $2

Next, let's calculate the expected profit per ticket. We can do this by multiplying the cost of each prize by its corresponding probability and summing them up:

Expected Profit per Ticket = (Probability of 1st Prize * Cost of 1st Prize) + (Probability of 2nd Prize * Cost of 2nd Prize) + (Probability of 3rd Prize * Cost of 3rd Prize)
= (0.15 * Cost of 1st Prize) + (0.30 * $1.28) + (0.45 * $0.70)

Since the expected profit per ticket is equal to the average expenses per ticket, we can set up the equation:

$2 = (0.15 * Cost of 1st Prize) + (0.30 * $1.28) + (0.45 * $0.70)

Now, let's solve for the Cost of 1st Prize:

$2 - (0.30 * $1.28) - (0.45 * $0.70) = 0.15 * Cost of 1st Prize
$2 - $0.384 - $0.315 = 0.15 * Cost of 1st Prize
$1.301 = 0.15 * Cost of 1st Prize

To find the Cost of the 1st Prize, we can divide both sides of the equation by 0.15:

Cost of 1st Prize = $1.301 / 0.15
≈ $8.67 (rounded to the nearest cent)

Therefore, Debra should spend approximately $8.67 on each first prize.