Answer :
To determine the probability that a customer will be seated either at a round table or by the window, we need to use the principle of inclusion-exclusion. Here's how you can solve it step-by-step:
1. Identify Total Items:
- Total number of tables in the restaurant: 60
2. Identify Sets:
- Number of round tables: 38
- Number of tables by the window: 13
- Number of round tables that are also by the window: 6
3. Apply Inclusion-Exclusion Principle:
The probability of a table being either round or by the window is given by the formula:
[tex]\[
P(\text{Round or Window}) = P(\text{Round}) + P(\text{Window}) - P(\text{Round and Window})
\][/tex]
- Probability of round tables: [tex]\( \frac{38}{60} \)[/tex]
- Probability of window tables: [tex]\( \frac{13}{60} \)[/tex]
- Probability of round tables by the window: [tex]\( \frac{6}{60} \)[/tex]
Substitute these values into the formula:
[tex]\[
P(\text{Round or Window}) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60}
\][/tex]
Simplify the expression:
[tex]\[
P(\text{Round or Window}) = \frac{38 + 13 - 6}{60} = \frac{45}{60}
\][/tex]
4. Simplify the Fraction:
- Simplifying [tex]\( \frac{45}{60} \)[/tex], we divide both the numerator and the denominator by their greatest common divisor, which is 15:
[tex]\[
\frac{45 \div 15}{60 \div 15} = \frac{3}{4}
\][/tex]
5. Convert Fraction to Decimal:
- The decimal form of [tex]\( \frac{3}{4} \)[/tex] is 0.75.
Therefore, the probability that a customer will be seated at a round table or by the window is [tex]\( \frac{45}{60} \)[/tex] or 0.75, which corresponds to option C: [tex]\(\frac{45}{60}\)[/tex].
1. Identify Total Items:
- Total number of tables in the restaurant: 60
2. Identify Sets:
- Number of round tables: 38
- Number of tables by the window: 13
- Number of round tables that are also by the window: 6
3. Apply Inclusion-Exclusion Principle:
The probability of a table being either round or by the window is given by the formula:
[tex]\[
P(\text{Round or Window}) = P(\text{Round}) + P(\text{Window}) - P(\text{Round and Window})
\][/tex]
- Probability of round tables: [tex]\( \frac{38}{60} \)[/tex]
- Probability of window tables: [tex]\( \frac{13}{60} \)[/tex]
- Probability of round tables by the window: [tex]\( \frac{6}{60} \)[/tex]
Substitute these values into the formula:
[tex]\[
P(\text{Round or Window}) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60}
\][/tex]
Simplify the expression:
[tex]\[
P(\text{Round or Window}) = \frac{38 + 13 - 6}{60} = \frac{45}{60}
\][/tex]
4. Simplify the Fraction:
- Simplifying [tex]\( \frac{45}{60} \)[/tex], we divide both the numerator and the denominator by their greatest common divisor, which is 15:
[tex]\[
\frac{45 \div 15}{60 \div 15} = \frac{3}{4}
\][/tex]
5. Convert Fraction to Decimal:
- The decimal form of [tex]\( \frac{3}{4} \)[/tex] is 0.75.
Therefore, the probability that a customer will be seated at a round table or by the window is [tex]\( \frac{45}{60} \)[/tex] or 0.75, which corresponds to option C: [tex]\(\frac{45}{60}\)[/tex].
