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 spent $1.54 on each second place prize and $0.77 on each third place prize. She wants the first prize to be nice enough to attract people to the game. The game's manufacturer has told her that the probability of winning a third prize is 0.41, of winning a second prize is 0.28 and of winning a first prize is 0.21. Tickets cost $3 each and she wants the school to profit an average of $ 1 per ticket.
What does the expected value of each ticket need to be in order for the school to make a profit of $ 1 per ticket? $
How much should she spend on each first prize? $
To find the expected value of each ticket, we need to calculate the average profit per ticket. Let's break down the calculations step by step:
1. Calculate the average profit from each prize category:
- For third place prize:
- Cost per prize = $0.77
- Probability of winning a third prize = 0.41
- Expected cost for winning a third prize = (Cost per prize) x (Probability of winning a third prize)
= $0.77 x 0.41 = $0.3177
- For second place prize:
- Cost per prize = $1.54
- Probability of winning a second prize = 0.28
- Expected cost for winning a second prize = (Cost per prize) x (Probability of winning a second prize)
= $1.54 x 0.28 = $0.4312
2. Calculate the average profit from the first prize:
- Let the desired average profit from each ticket be x dollars.
- Probability of winning a first prize = 0.21
- Expected cost for winning a first prize = (Cost per prize) x (Probability of winning a first prize) = $x x 0.21
3. Determine the average profit from a ticket:
- Let the ticket price be $3.
- Average profit from a ticket = Ticket price - (Expected cost for winning a third prize + Expected cost for winning a second prize + Expected cost for winning a first prize)
= $3 - ($0.3177 + $0.4312 + $x x 0.21)
4. Set up the equation to find x:
- We want the school to make a profit of $1 per ticket. Therefore, the average profit from a ticket should be $1.
- Set Average profit from a ticket = $1 and solve for x:
$3 - ($0.3177 + $0.4312 + $x x 0.21) = $1
Solving the equation will give us the value of x, which represents the desired average profit per ticket.
To calculate how much she should spend on each first prize, we need to substitute the value of x into the equation for the expected cost of winning a first prize:
Expected cost for winning a first prize = $x x 0.21
With the value of x, we can calculate the desired expenditure for each first prize.
To find the expected value of each ticket, we need to consider the probabilities of winning each prize level and the cost of each ticket.
Let's define:
P1 = Probability of winning a first prize
P2 = Probability of winning a second prize
P3 = Probability of winning a third prize
C = Cost of each ticket
We know that the ticket cost is $3 and the school wants to make a profit of $1 per ticket. Therefore, the expected value of each ticket should be $4 ($3 ticket cost + $1 profit).
To calculate the expected value, we multiply the probability of winning each prize level by the respective prize value and sum them up. Thus, we have the following equation:
(4 * P1) + (2 * P2) + (1 * P3) = 4
Now, let's substitute the given probabilities into the equation:
(4 * P1) + (2 * 0.28) + (1 * 0.41) = 4
Simplifying the equation:
4P1 + 0.56 + 0.41 = 4
4P1 + 0.97 = 4
4P1 = 4 - 0.97
4P1 = 3.03
P1 = 3.03 / 4
P1 = 0.7575
Therefore, the probability of winning a first prize is 0.7575.
Now, let's calculate how much Debra should spend on each first prize.
Assuming Debra spends X dollars on each first prize, the expected value of the first prize will also be X.
Using the same equation as before, we can calculate the expected value:
(4 * 0.7575) + (2 * 0.28) + (1 * 0.41) = 4X
3.03 + 0.56 + 0.41 = 4X
4.00 = 4X
Dividing both sides of the equation by 4:
X = 4.00 / 4
X = 1.00
Therefore, Debra should spend $1 on each first prize.