Answer :
- Define events: R (round table), W (table by window).
- Calculate individual probabilities: $P(R) = \frac{38}{60}$, $P(W) = \frac{13}{60}$, $P(R \cap W) = \frac{6}{60}$.
- Apply the formula for the union of two events: $P(R \cup W) = P(R) + P(W) - P(R \cap W)$.
- Calculate the final probability: $P(R \cup W) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60} = \boxed{\frac{45}{60}}$.
### Explanation
1. Understand the problem
Let's analyze the problem. We are given the number of round tables, the number of tables by the window, and the number of round tables by the window in a restaurant. We want to find the probability that a customer will be seated at a round table or by the window.
2. Define events and probabilities
Let R be the event that a table is round, and W be the event that a table is by the window. We are given:
- Total number of tables = 60
- Number of round tables = 38, so $P(R) = \frac{38}{60}$
- Number of tables by the window = 13, so $P(W) = \frac{13}{60}$
- Number of round tables by the window = 6, so $P(R \cap W) = \frac{6}{60}$
3. Apply the formula and calculate
We want to find $P(R \cup W)$, which is the probability that a table is round or by the window (or both). We can use the formula for the union of two events:
$$P(R \cup W) = P(R) + P(W) - P(R \cap W)$$
Substituting the given values, we have:
$$P(R \cup W) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60}$$
$$P(R \cup W) = \frac{38 + 13 - 6}{60} = \frac{45}{60}$$
Simplifying the fraction, we get:
$$P(R \cup W) = \frac{45}{60} = \frac{3}{4} = 0.75$$
4. State the final answer
The probability that a customer will be seated at a round table or by the window is $\frac{45}{60}$, which simplifies to $\frac{3}{4}$. Looking at the answer choices, we see that option C is $\frac{45}{60}$.
### Examples
This type of probability calculation is useful in many real-world scenarios. For example, a marketing team might want to know the probability that a customer likes either product A or product B. Or a city planner might want to know the probability that a resident uses either public transportation or a bicycle to commute. Understanding how to calculate the probability of the union of two events is a valuable skill in many fields.
- Calculate individual probabilities: $P(R) = \frac{38}{60}$, $P(W) = \frac{13}{60}$, $P(R \cap W) = \frac{6}{60}$.
- Apply the formula for the union of two events: $P(R \cup W) = P(R) + P(W) - P(R \cap W)$.
- Calculate the final probability: $P(R \cup W) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60} = \boxed{\frac{45}{60}}$.
### Explanation
1. Understand the problem
Let's analyze the problem. We are given the number of round tables, the number of tables by the window, and the number of round tables by the window in a restaurant. We want to find the probability that a customer will be seated at a round table or by the window.
2. Define events and probabilities
Let R be the event that a table is round, and W be the event that a table is by the window. We are given:
- Total number of tables = 60
- Number of round tables = 38, so $P(R) = \frac{38}{60}$
- Number of tables by the window = 13, so $P(W) = \frac{13}{60}$
- Number of round tables by the window = 6, so $P(R \cap W) = \frac{6}{60}$
3. Apply the formula and calculate
We want to find $P(R \cup W)$, which is the probability that a table is round or by the window (or both). We can use the formula for the union of two events:
$$P(R \cup W) = P(R) + P(W) - P(R \cap W)$$
Substituting the given values, we have:
$$P(R \cup W) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60}$$
$$P(R \cup W) = \frac{38 + 13 - 6}{60} = \frac{45}{60}$$
Simplifying the fraction, we get:
$$P(R \cup W) = \frac{45}{60} = \frac{3}{4} = 0.75$$
4. State the final answer
The probability that a customer will be seated at a round table or by the window is $\frac{45}{60}$, which simplifies to $\frac{3}{4}$. Looking at the answer choices, we see that option C is $\frac{45}{60}$.
### Examples
This type of probability calculation is useful in many real-world scenarios. For example, a marketing team might want to know the probability that a customer likes either product A or product B. Or a city planner might want to know the probability that a resident uses either public transportation or a bicycle to commute. Understanding how to calculate the probability of the union of two events is a valuable skill in many fields.
