Question

The management of a hotel conducted a survey. It found that of the 2560 guests who were surveyed, 1785 tip the wait staff. 1219 tip the luggage handlers. 831 tip the maids. 275 tip the maids and the luggage handlers. 700 tip the wait staff and the maids. 755 tip the wait staff and the luggage handlers. 245 tip all three services. 210 do not tip these services. How many of the surveyed guests tip a. exactly two of the three services

Answer 1

To find out how many of the surveyed guests tip exactly two of the three services, we can use the principle of inclusion-exclusion.

Let's define some sets:

A = Guests who tip the wait staff.

B = Guests who tip the luggage handlers.

C = Guests who tip the maids.

We are given the following information:

- |A| = 1785 (number of guests who tip the wait staff).

- |B| = 1219 (number of guests who tip the luggage handlers).

- |C| = 831 (number of guests who tip the maids).

- |A ∩ B| = 755 (number of guests who tip both the wait staff and luggage handlers).

- |A ∩ C| = 700 (number of guests who tip both the wait staff and maids).

- |B ∩ C| = 275 (number of guests who tip both luggage handlers and maids).

- |A ∩ B ∩ C| = 245 (number of guests who tip all three services).

- |Total Surveyed Guests| = 2560.

- |No Tip| = 210 (number of guests who do not tip any service).

We want to find out how many guests tip exactly two of the three services, which means they don't tip the third service. We can use the principle of inclusion-exclusion to calculate this:

Number of guests who tip exactly two services = |(A ∩ B) ∪ (A ∩ C) ∪ (B ∩ C)|

Using the principle of inclusion-exclusion:

|A ∩ B ∪ A ∩ C ∪ B ∩ C| = |A ∩ B| + |A ∩ C| + |B ∩ C| - 2|A ∩ B ∩ C|

Now, plug in the given values:

|A ∩ B ∪ A ∩ C ∪ B ∩ C| = 755 + 700 + 275 - 2 * 245

|A ∩ B ∪ A ∩ C ∪ B ∩ C| = 1730 - 490

|A ∩ B ∪ A ∩ C ∪ B ∩ C| = 1240

So, 1240 of the surveyed guests tip exactly two of the three services.