A travel agency offers 4 different vacation packages to Europe. Their net profit for package 1 is $500, for package 2 it is $750, for package 3 it is $900, and for package 4 it is $1,500. From past experience they know that 20% of their customers purchase package 1, 15% of their customers purchase package 2, 40% of their customers purchase package 3 and 25% of their customers purchase package 4. Find the expected value or average profit per customer and determine how much profit they should expect if 10 people purchase one of their European vacation packages

To find the expected value or average profit per customer, we need to calculate the weighted average of the profits of each package.

First, let's calculate the expected profit for each package:
- Expected profit for package 1 = 20% * $500 = $100
- Expected profit for package 2 = 15% * $750 = $112.50
- Expected profit for package 3 = 40% * $900 = $360
- Expected profit for package 4 = 25% * $1500 = $375

Now, we can calculate the overall expected value by taking the weighted average of these profits:

Expected value = (20% * $100) + (15% * $112.50) + (40% * $360) + (25% * $375)
Expected value = $20 + $16.88 + $144 + $93.75
Expected value = $274.63

Therefore, the expected average profit per customer is $274.63.

To determine the expected profit if 10 people purchase one of their European vacation packages, simply multiply the expected value by the number of customers:

Expected profit = $274.63 * 10
Expected profit = $2,746.30

Therefore, the travel agency should expect a profit of $2,746.30 if 10 people purchase one of their European vacation packages.

To find the expected value or average profit per customer, we need to multiply the profit for each package by the corresponding probability and then sum up these values.

Step 1: Calculate the expected profit for each package:
Package 1: $500 * 20% = $100
Package 2: $750 * 15% = $112.50
Package 3: $900 * 40% = $360
Package 4: $1,500 * 25% = $375

Step 2: Calculate the total expected profit:
Total Expected Profit = $100 + $112.50 + $360 + $375 = $947.50

Step 3: Calculate the average profit per customer:
Average Profit per Customer = Total Expected Profit / Number of Customers
Here, it is given that there are 10 customers, so:
Average Profit per Customer = $947.50 / 10 = $94.75

Therefore, the average profit per customer or the expected value is $94.75.

Now, to determine the total profit they should expect if 10 people purchase one of their European vacation packages, we can simply multiply the average profit per customer by the number of customers:
Total Profit = Average Profit per Customer * Number of Customers
Total Profit = $94.75 * 10 = $947.50

Therefore, they should expect a profit of $947.50 if 10 people purchase one of their European vacation packages.