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Answer :
To solve the problem of finding the probability that a customer will be seated at either a round table or a table by the window, we will apply the principle of inclusion-exclusion. Here is a step-by-step outline of the process:
1. Information Provided:
- Total number of tables, [tex]\( T = 60 \)[/tex]
- Number of round tables, [tex]\( R = 38 \)[/tex]
- Number of tables by the window, [tex]\( W = 13 \)[/tex]
- Number of round tables by the window, [tex]\( RW = 6 \)[/tex]
2. Identify the Required Probability:
We need to find the probability [tex]\( P(R \cup W) \)[/tex] where a table is either round or by the window.
3. Inclusion-Exclusion Principle:
Using the principle of inclusion-exclusion for counting elements in the union of two sets, we have:
[tex]\[
P(R \cup W) = P(R) + P(W) - P(R \cap W)
\][/tex]
- [tex]\( P(R) \)[/tex] is the probability of a round table:
[tex]\[
P(R) = \frac{R}{T} = \frac{38}{60}
\][/tex]
- [tex]\( P(W) \)[/tex] is the probability of a table by the window:
[tex]\[
P(W) = \frac{W}{T} = \frac{13}{60}
\][/tex]
- [tex]\( P(R \cap W) \)[/tex] is the probability of a round table by the window:
[tex]\[
P(R \cap W) = \frac{RW}{T} = \frac{6}{60}
\][/tex]
4. Calculate the combined probability:
Substitute these values into the principle of inclusion-exclusion formula:
[tex]\[
P(R \cup W) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60}
\][/tex]
5. Simplify the Expression:
[tex]\[
P(R \cup W) = \frac{38 + 13 - 6}{60}
\][/tex]
[tex]\[
P(R \cup W) = \frac{45}{60}
\][/tex]
6. Convert the Probability to Fraction and Simplify:
Simplify [tex]\(\frac{45}{60}\)[/tex]:
[tex]\[
\frac{45}{60} = \frac{3}{4}
\][/tex]
7. Verify and Match the Answer with Choices:
The simplified fraction [tex]\(\frac{3}{4}\)[/tex] corresponds to the given probabilities. Only one of the answer choices matches:
[tex]\[
\frac{3}{4} = \frac{45}{60}, \text{ Answer B}
\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{45}{60}} \][/tex]
1. Information Provided:
- Total number of tables, [tex]\( T = 60 \)[/tex]
- Number of round tables, [tex]\( R = 38 \)[/tex]
- Number of tables by the window, [tex]\( W = 13 \)[/tex]
- Number of round tables by the window, [tex]\( RW = 6 \)[/tex]
2. Identify the Required Probability:
We need to find the probability [tex]\( P(R \cup W) \)[/tex] where a table is either round or by the window.
3. Inclusion-Exclusion Principle:
Using the principle of inclusion-exclusion for counting elements in the union of two sets, we have:
[tex]\[
P(R \cup W) = P(R) + P(W) - P(R \cap W)
\][/tex]
- [tex]\( P(R) \)[/tex] is the probability of a round table:
[tex]\[
P(R) = \frac{R}{T} = \frac{38}{60}
\][/tex]
- [tex]\( P(W) \)[/tex] is the probability of a table by the window:
[tex]\[
P(W) = \frac{W}{T} = \frac{13}{60}
\][/tex]
- [tex]\( P(R \cap W) \)[/tex] is the probability of a round table by the window:
[tex]\[
P(R \cap W) = \frac{RW}{T} = \frac{6}{60}
\][/tex]
4. Calculate the combined probability:
Substitute these values into the principle of inclusion-exclusion formula:
[tex]\[
P(R \cup W) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60}
\][/tex]
5. Simplify the Expression:
[tex]\[
P(R \cup W) = \frac{38 + 13 - 6}{60}
\][/tex]
[tex]\[
P(R \cup W) = \frac{45}{60}
\][/tex]
6. Convert the Probability to Fraction and Simplify:
Simplify [tex]\(\frac{45}{60}\)[/tex]:
[tex]\[
\frac{45}{60} = \frac{3}{4}
\][/tex]
7. Verify and Match the Answer with Choices:
The simplified fraction [tex]\(\frac{3}{4}\)[/tex] corresponds to the given probabilities. Only one of the answer choices matches:
[tex]\[
\frac{3}{4} = \frac{45}{60}, \text{ Answer B}
\][/tex]
Therefore, the correct answer is:
[tex]\[ \boxed{\frac{45}{60}} \][/tex]
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