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Answer :
To solve this problem, we need 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 to find the solution.
Let's break it down step-by-step:
1. Identify the Information Given:
- Total number of tables = 60
- Number of round tables = 38
- Number of tables by the window = 13
- Number of round tables by the window = 6
2. Determine What We Need:
- We need the probability that a customer is seated at a round table or a table by the window.
3. Apply the Inclusion-Exclusion Principle:
- We want to find [tex]\( P(\text{round or window}) \)[/tex].
- The formula for this using the principle of inclusion-exclusion is:
[tex]\[
P(\text{round or window}) = P(\text{round}) + P(\text{window}) - P(\text{round and window})
\][/tex]
4. Calculate the Numbers:
- [tex]\( P(\text{round}) \)[/tex] refers to the number of round tables, which is 38.
- [tex]\( P(\text{window}) \)[/tex] refers to the number of tables by the window, which is 13.
- [tex]\( P(\text{round and window}) \)[/tex] refers to the number of tables that are both round and by the window, which is 6.
5. Calculate the Total Favorable Outcomes:
- Add the round tables and window tables, and then subtract the overlap (round and window) to avoid double-counting:
[tex]\[
\text{Favorables} = 38 + 13 - 6 = 45
\][/tex]
6. Calculate the Probability:
- Divide the number of favorable outcomes by the total number of tables:
[tex]\[
\text{Probability} = \frac{45}{60}
\][/tex]
7. Determine the Correct Answer:
- The probability simplifies to [tex]\(\frac{45}{60}\)[/tex].
- Comparing this with the given options, the answer is option A: [tex]\(\frac{45}{60}\)[/tex].
So, the probability that a customer will be seated at a round table or a table by the window is [tex]\(\frac{45}{60}\)[/tex], which matches answer choice A.
Let's break it down step-by-step:
1. Identify the Information Given:
- Total number of tables = 60
- Number of round tables = 38
- Number of tables by the window = 13
- Number of round tables by the window = 6
2. Determine What We Need:
- We need the probability that a customer is seated at a round table or a table by the window.
3. Apply the Inclusion-Exclusion Principle:
- We want to find [tex]\( P(\text{round or window}) \)[/tex].
- The formula for this using the principle of inclusion-exclusion is:
[tex]\[
P(\text{round or window}) = P(\text{round}) + P(\text{window}) - P(\text{round and window})
\][/tex]
4. Calculate the Numbers:
- [tex]\( P(\text{round}) \)[/tex] refers to the number of round tables, which is 38.
- [tex]\( P(\text{window}) \)[/tex] refers to the number of tables by the window, which is 13.
- [tex]\( P(\text{round and window}) \)[/tex] refers to the number of tables that are both round and by the window, which is 6.
5. Calculate the Total Favorable Outcomes:
- Add the round tables and window tables, and then subtract the overlap (round and window) to avoid double-counting:
[tex]\[
\text{Favorables} = 38 + 13 - 6 = 45
\][/tex]
6. Calculate the Probability:
- Divide the number of favorable outcomes by the total number of tables:
[tex]\[
\text{Probability} = \frac{45}{60}
\][/tex]
7. Determine the Correct Answer:
- The probability simplifies to [tex]\(\frac{45}{60}\)[/tex].
- Comparing this with the given options, the answer is option A: [tex]\(\frac{45}{60}\)[/tex].
So, the probability that a customer will be seated at a round table or a table by the window is [tex]\(\frac{45}{60}\)[/tex], which matches answer choice A.
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