Answer :
To solve the problem of finding the probability that a customer will be seated at a round table or by the window, we can use the formula for the probability of the union of two events.
Let's break this down step-by-step:
1. Identify the Events:
- Event A: A table is round.
- Event B: A table is by the window.
2. Given Values:
- Total tables = 60
- Round tables = 38
- Tables by the window = 13
- Round tables by the window = 6
3. Apply the Formula:
- The probability of the union of two events (A or B) is given by:
[tex]\[
P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
\][/tex]
- [tex]\( P(A) \)[/tex] is the probability of a round table, which is the number of round tables divided by total tables. So:
[tex]\[
P(A) = \frac{38}{60}
\][/tex]
- [tex]\( P(B) \)[/tex] is the probability of a table by the window, which is the number of window tables divided by total tables. So:
[tex]\[
P(B) = \frac{13}{60}
\][/tex]
- [tex]\( P(A \text{ and } B) \)[/tex] is the probability of both round and by the window, which is the number of round tables by the window divided by total tables. So:
[tex]\[
P(A \text{ and } B) = \frac{6}{60}
\][/tex]
4. Calculate the Probability:
- Substituting these values into the formula:
[tex]\[
P(A \text{ or } B) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60}
\][/tex]
- This simplifies to:
[tex]\[
P(A \text{ or } B) = \frac{45}{60} = 0.75
\][/tex]
5. Convert to a Fraction:
- The probability expressed as a fraction from the total number of tables is:
[tex]\[
P(A \text{ or } B) = \frac{45}{60}
\][/tex]
Given these steps, the correct answer is option A: [tex]\(\frac{45}{60}\)[/tex].
Let's break this down step-by-step:
1. Identify the Events:
- Event A: A table is round.
- Event B: A table is by the window.
2. Given Values:
- Total tables = 60
- Round tables = 38
- Tables by the window = 13
- Round tables by the window = 6
3. Apply the Formula:
- The probability of the union of two events (A or B) is given by:
[tex]\[
P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)
\][/tex]
- [tex]\( P(A) \)[/tex] is the probability of a round table, which is the number of round tables divided by total tables. So:
[tex]\[
P(A) = \frac{38}{60}
\][/tex]
- [tex]\( P(B) \)[/tex] is the probability of a table by the window, which is the number of window tables divided by total tables. So:
[tex]\[
P(B) = \frac{13}{60}
\][/tex]
- [tex]\( P(A \text{ and } B) \)[/tex] is the probability of both round and by the window, which is the number of round tables by the window divided by total tables. So:
[tex]\[
P(A \text{ and } B) = \frac{6}{60}
\][/tex]
4. Calculate the Probability:
- Substituting these values into the formula:
[tex]\[
P(A \text{ or } B) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60}
\][/tex]
- This simplifies to:
[tex]\[
P(A \text{ or } B) = \frac{45}{60} = 0.75
\][/tex]
5. Convert to a Fraction:
- The probability expressed as a fraction from the total number of tables is:
[tex]\[
P(A \text{ or } B) = \frac{45}{60}
\][/tex]
Given these steps, the correct answer is option A: [tex]\(\frac{45}{60}\)[/tex].
