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Answer :
To determine 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 to avoid double-counting the tables that are both round and by the window.
Let's break down the problem step-by-step:
1. Total number of tables:
There are 60 tables in the restaurant.
2. Number of round tables:
There are 38 round tables.
3. Number of tables by the window:
There are 13 tables by the window.
4. Number of tables that are both round and by the window:
There are 6 tables that are both round and by the window.
5. Using the principle of inclusion-exclusion:
The principle of inclusion-exclusion states that to find the total number of tables that are either round or by the window, we need to add the number of round tables to the number of tables by the window and then subtract the number of tables that are both round and by the window.
[tex]\[
\text{Tables either round or by the window} = \text{Number of round tables} + \text{Number of window tables} - \text{Number of tables that are both round and by the window}
\][/tex]
Plugging in the values:
[tex]\[
\text{Tables either round or by the window} = 38 + 13 - 6 = 45
\][/tex]
6. Probability calculation:
The probability is calculated as the number of favorable outcomes (tables either round or by the window) divided by the total number of possible outcomes (total tables).
[tex]\[
\text{Probability} = \frac{\text{Tables either round or by the window}}{\text{Total tables}} = \frac{45}{60}
\][/tex]
7. Simplifying the fraction:
The fraction [tex]\(\frac{45}{60}\)[/tex] can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 15.
[tex]\[
\frac{45}{60} = \frac{45 \div 15}{60 \div 15} = \frac{3}{4}
\][/tex]
Hence, 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.
Given the options:
- A. [tex]\(\frac{45}{60}\)[/tex]
- B. [tex]\(\frac{41}{60}\)[/tex]
- C. [tex]\(\frac{29}{60}\)[/tex]
- D. [tex]\(\frac{47}{60}\)[/tex]
The correct answer is:
A. [tex]\(\frac{45}{60}\)[/tex]
Let's break down the problem step-by-step:
1. Total number of tables:
There are 60 tables in the restaurant.
2. Number of round tables:
There are 38 round tables.
3. Number of tables by the window:
There are 13 tables by the window.
4. Number of tables that are both round and by the window:
There are 6 tables that are both round and by the window.
5. Using the principle of inclusion-exclusion:
The principle of inclusion-exclusion states that to find the total number of tables that are either round or by the window, we need to add the number of round tables to the number of tables by the window and then subtract the number of tables that are both round and by the window.
[tex]\[
\text{Tables either round or by the window} = \text{Number of round tables} + \text{Number of window tables} - \text{Number of tables that are both round and by the window}
\][/tex]
Plugging in the values:
[tex]\[
\text{Tables either round or by the window} = 38 + 13 - 6 = 45
\][/tex]
6. Probability calculation:
The probability is calculated as the number of favorable outcomes (tables either round or by the window) divided by the total number of possible outcomes (total tables).
[tex]\[
\text{Probability} = \frac{\text{Tables either round or by the window}}{\text{Total tables}} = \frac{45}{60}
\][/tex]
7. Simplifying the fraction:
The fraction [tex]\(\frac{45}{60}\)[/tex] can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 15.
[tex]\[
\frac{45}{60} = \frac{45 \div 15}{60 \div 15} = \frac{3}{4}
\][/tex]
Hence, 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.
Given the options:
- A. [tex]\(\frac{45}{60}\)[/tex]
- B. [tex]\(\frac{41}{60}\)[/tex]
- C. [tex]\(\frac{29}{60}\)[/tex]
- D. [tex]\(\frac{47}{60}\)[/tex]
The correct answer is:
A. [tex]\(\frac{45}{60}\)[/tex]
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