ent-4-wheels makes a profit of $12 per day for each car rented, and loses $4 per day for each car that is not rented. How many cars should the company have on hand to maximize profit, subject to the following probability distribution:
anticipated number of rental requests
12 11 10 9 8 7 6
Probability
.09 .011 .22 .24 .12 .12 .10
This is what I did and I;m not sure if it's correct.
(12)(.09)+(4)(.91)=4.72
(11)(.11)+(4)(.89)=4.77
ect. i got that company should have ten cars to maximize profit
To maximize profit, you need to find the number of cars that will result in the highest overall profit based on the probability distribution provided.
First, let's calculate the profit for each possible number of rental requests:
For 12 rental requests:
Profit = (12 * 12) - (4 * (10 - 12)) = $120
For 11 rental requests:
Profit = (12 * 11) - (4 * (10 - 11)) = $116
For 10 rental requests:
Profit = (12 * 10) - (4 * (10 - 10)) = $120
For 9 rental requests:
Profit = (12 * 9) - (4 * (10 - 9)) = $116
For 8 rental requests:
Profit = (12 * 8) - (4 * (10 - 8)) = $112
For 7 rental requests:
Profit = (12 * 7) - (4 * (10 - 7)) = $100
For 6 rental requests:
Profit = (12 * 6) - (4 * (10 - 6)) = $88
Now, let's calculate the expected profit by multiplying each profit by its respective probability:
Expected Profit for 12 rental requests = $120 * 0.09 = $10.80
Expected Profit for 11 rental requests = $116 * 0.011 = $1.28
Expected Profit for 10 rental requests = $120 * 0.22 = $26.40
Expected Profit for 9 rental requests = $116 * 0.24 = $27.84
Expected Profit for 8 rental requests = $112 * 0.12 = $13.44
Expected Profit for 7 rental requests = $100 * 0.12 = $12.00
Expected Profit for 6 rental requests = $88 * 0.10 = $8.80
Now, sum up the expected profits for each scenario:
Expected Total Profit = $10.80 + $1.28 + $26.40 + $27.84 + $13.44 + $12.00 + $8.80 = $100.56
Based on this analysis, it can be concluded that the company should have 10 cars on hand to maximize profit, as it results in the highest expected total profit.
So, your calculation of having 10 cars to maximize profit is correct.