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Answer :
To solve this problem, we want to find the probability that a customer will be seated at a round table or a table by the window.
Here’s a step-by-step explanation:
1. Understand the Sets:
- Total number of tables in the restaurant: 60
- Number of round tables: 38
- Number of tables by the window: 13
- Number of tables that are both round and by the window: 6
2. Using the Principle of Inclusion-Exclusion:
The probability of seating at a round table or by a window is calculated using the principle of inclusion-exclusion. This principle helps us avoid double-counting tables that are both round and by the window.
The formula is:
[tex]\[
P(\text{Round or Window}) = P(\text{Round}) + P(\text{Window}) - P(\text{Round and Window})
\][/tex]
3. Calculate the Individual Probabilities:
- The number of round tables is 38.
- The number of window tables is 13.
- The number of tables that are both round and by the window is 6.
4. Apply the Formula:
Substitute these values into the formula. The number of favorable outcomes for being either a round table or a window table is calculated as follows:
[tex]\[
\text{Round or Window} = 38 + 13 - 6 = 45
\][/tex]
The total number of tables is 60. So, the probability is:
[tex]\[
\text{Probability} = \frac{45}{60}
\][/tex]
5. Calculate the Probability:
Simplify [tex]\(\frac{45}{60}\)[/tex] to [tex]\(\frac{3}{4}\)[/tex].
6. Convert to Decimal:
[tex]\(\frac{3}{4}\)[/tex] is 0.75 as a decimal.
Therefore, the probability that a customer will be seated at a round table or by the window is 0.75, which simplifies to [tex]\(\frac{45}{60}\)[/tex].
Looking at the given options, the correct answer is:
- B. [tex]\(\frac{45}{60}\)[/tex].
Here’s a step-by-step explanation:
1. Understand the Sets:
- Total number of tables in the restaurant: 60
- Number of round tables: 38
- Number of tables by the window: 13
- Number of tables that are both round and by the window: 6
2. Using the Principle of Inclusion-Exclusion:
The probability of seating at a round table or by a window is calculated using the principle of inclusion-exclusion. This principle helps us avoid double-counting tables that are both round and by the window.
The formula is:
[tex]\[
P(\text{Round or Window}) = P(\text{Round}) + P(\text{Window}) - P(\text{Round and Window})
\][/tex]
3. Calculate the Individual Probabilities:
- The number of round tables is 38.
- The number of window tables is 13.
- The number of tables that are both round and by the window is 6.
4. Apply the Formula:
Substitute these values into the formula. The number of favorable outcomes for being either a round table or a window table is calculated as follows:
[tex]\[
\text{Round or Window} = 38 + 13 - 6 = 45
\][/tex]
The total number of tables is 60. So, the probability is:
[tex]\[
\text{Probability} = \frac{45}{60}
\][/tex]
5. Calculate the Probability:
Simplify [tex]\(\frac{45}{60}\)[/tex] to [tex]\(\frac{3}{4}\)[/tex].
6. Convert to Decimal:
[tex]\(\frac{3}{4}\)[/tex] is 0.75 as a decimal.
Therefore, the probability that a customer will be seated at a round table or by the window is 0.75, which simplifies to [tex]\(\frac{45}{60}\)[/tex].
Looking at the given options, the correct answer is:
- B. [tex]\(\frac{45}{60}\)[/tex].
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