Answer :
To find the probability that a customer will be seated at a round table or by the window, we can use the principle of inclusion-exclusion. This principle helps us avoid double-counting when we have overlapping groups.
Here's a step-by-step breakdown:
1. Identify the total number of tables:
The restaurant has a total of 60 tables.
2. Identify the number of round tables:
There are 38 round tables.
3. Identify the number of tables by the window:
There are 13 tables located by the window.
4. Identify the number of tables that are both round and by the window:
There are 6 tables that are both round and by the window.
5. Apply the principle of inclusion-exclusion:
To find the total number of tables that are either round or by the window, we add the number of round tables and the number of tables by the window, then subtract the number of tables that are both round and by the window (since these tables are counted twice):
[tex]\[
\text{Round or window tables} = (\text{Round tables}) + (\text{Window tables}) - (\text{Round and window tables})
\][/tex]
[tex]\[
= 38 + 13 - 6 = 45
\][/tex]
6. Calculate the probability:
Probability is the number of favorable outcomes (round or window tables) divided by the total number of outcomes (total tables):
[tex]\[
\text{Probability} = \frac{\text{Round or window tables}}{\text{Total tables}} = \frac{45}{60}
\][/tex]
7. Simplify the fraction:
The fraction [tex]\(\frac{45}{60}\)[/tex] simplifies to [tex]\(\frac{3}{4}\)[/tex], which is equivalent to a probability of 0.75.
Therefore, the probability that a customer will be seated at a round table or by the window is [tex]\( \frac{3}{4} \)[/tex] or 0.75. This corresponds to option C, [tex]\( \frac{45}{60} \)[/tex].
Here's a step-by-step breakdown:
1. Identify the total number of tables:
The restaurant has a total of 60 tables.
2. Identify the number of round tables:
There are 38 round tables.
3. Identify the number of tables by the window:
There are 13 tables located by the window.
4. Identify the number of tables that are both round and by the window:
There are 6 tables that are both round and by the window.
5. Apply the principle of inclusion-exclusion:
To find the total number of tables that are either round or by the window, we add the number of round tables and the number of tables by the window, then subtract the number of tables that are both round and by the window (since these tables are counted twice):
[tex]\[
\text{Round or window tables} = (\text{Round tables}) + (\text{Window tables}) - (\text{Round and window tables})
\][/tex]
[tex]\[
= 38 + 13 - 6 = 45
\][/tex]
6. Calculate the probability:
Probability is the number of favorable outcomes (round or window tables) divided by the total number of outcomes (total tables):
[tex]\[
\text{Probability} = \frac{\text{Round or window tables}}{\text{Total tables}} = \frac{45}{60}
\][/tex]
7. Simplify the fraction:
The fraction [tex]\(\frac{45}{60}\)[/tex] simplifies to [tex]\(\frac{3}{4}\)[/tex], which is equivalent to a probability of 0.75.
Therefore, the probability that a customer will be seated at a round table or by the window is [tex]\( \frac{3}{4} \)[/tex] or 0.75. This corresponds to option C, [tex]\( \frac{45}{60} \)[/tex].
