Answer :
To find the probability that a customer will be seated at a round table or by the window, we use the principle of Inclusion-Exclusion. Here's a step-by-step solution:
1. Identify the Total Number of Tables:
The restaurant has a total of 60 tables.
2. Identify Specific Groups of Tables:
- There are 38 round tables.
- There are 13 tables located by the window.
- There are 6 tables that are both round and located by the window.
3. Use the Principle of Inclusion-Exclusion:
We need to calculate the total number of tables that are either round or by the window. According to the Inclusion-Exclusion Principle, we can find this by adding the number of round tables to the number of window tables, then subtracting the number of tables that are counted twice (round tables by the window):
[tex]\[
\text{Tables that are round or by the window} = \text{round tables} + \text{window tables} - \text{round and window tables}
\][/tex]
Substitute the values we know:
[tex]\[
38 + 13 - 6 = 45
\][/tex]
So, there are 45 tables that are either round or located by the window.
4. Calculate the Probability:
To find the probability that a customer will be seated at one of these 45 tables, divide the number of favorable outcomes (tables that are round or by the window) by the total number of tables:
[tex]\[
\text{Probability} = \frac{\text{Number of tables round or by the window}}{\text{Total number of tables}} = \frac{45}{60}
\][/tex]
5. Simplify the Probability:
Simplify the fraction:
[tex]\[
\frac{45}{60} = \frac{3}{4} = 0.75
\][/tex]
Therefore, the probability that a customer will be seated at a round table or by the window is [tex]\(\frac{45}{60}\)[/tex], which simplifies to [tex]\(\frac{3}{4}\)[/tex] or 0.75. Thus, the correct answer is option C: [tex]\(\frac{45}{60}\)[/tex].
1. Identify the Total Number of Tables:
The restaurant has a total of 60 tables.
2. Identify Specific Groups of Tables:
- There are 38 round tables.
- There are 13 tables located by the window.
- There are 6 tables that are both round and located by the window.
3. Use the Principle of Inclusion-Exclusion:
We need to calculate the total number of tables that are either round or by the window. According to the Inclusion-Exclusion Principle, we can find this by adding the number of round tables to the number of window tables, then subtracting the number of tables that are counted twice (round tables by the window):
[tex]\[
\text{Tables that are round or by the window} = \text{round tables} + \text{window tables} - \text{round and window tables}
\][/tex]
Substitute the values we know:
[tex]\[
38 + 13 - 6 = 45
\][/tex]
So, there are 45 tables that are either round or located by the window.
4. Calculate the Probability:
To find the probability that a customer will be seated at one of these 45 tables, divide the number of favorable outcomes (tables that are round or by the window) by the total number of tables:
[tex]\[
\text{Probability} = \frac{\text{Number of tables round or by the window}}{\text{Total number of tables}} = \frac{45}{60}
\][/tex]
5. Simplify the Probability:
Simplify the fraction:
[tex]\[
\frac{45}{60} = \frac{3}{4} = 0.75
\][/tex]
Therefore, the probability that a customer will be seated at a round table or by the window is [tex]\(\frac{45}{60}\)[/tex], which simplifies to [tex]\(\frac{3}{4}\)[/tex] or 0.75. Thus, the correct answer is option C: [tex]\(\frac{45}{60}\)[/tex].
