Answer :
- Define events: A (round table), B (table by the window).
- Use inclusion-exclusion principle: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
- Calculate probabilities: $P(A) = \frac{38}{60}$, $P(B) = \frac{13}{60}$, $P(A \cap B) = \frac{6}{60}$.
- Find the final probability: $P(A \cup B) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60} = \boxed{\frac{45}{60}}$.
### Explanation
1. Understanding the Problem
Let's break down this probability problem step by step! We need to find the probability that a customer will be seated at a round table or a table by the window. To do this, we'll use the principle of inclusion-exclusion.
2. Defining Events
Let A be the event that a table is round, and B be the event that a table is by the window. We want to find $P(A \cup B)$, which means the probability of A or B happening.
3. Applying the Inclusion-Exclusion Principle
The formula for $P(A \cup B)$ is: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ This formula helps us avoid counting the tables that are both round and by the window twice.
4. Calculating Individual Probabilities
Now, let's calculate each probability:
$P(A)$ (probability of a round table) = $\frac{\text{Number of round tables}}{\text{Total number of tables}} = \frac{38}{60}$
$P(B)$ (probability of a table by the window) = $\frac{\text{Number of tables by the window}}{\text{Total number of tables}} = \frac{13}{60}$
$P(A \cap B)$ (probability of a round table by the window) = $\frac{\text{Number of round tables by the window}}{\text{Total number of tables}} = \frac{6}{60}$
5. Combining the Probabilities
Plug these values into the formula: $$P(A \cup B) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60}$$
6. Simplifying the Result
Now, let's simplify: $$P(A \cup B) = \frac{38 + 13 - 6}{60} = \frac{45}{60}$$
7. Final Answer
So, the probability that a customer will be seated at a round table or by the window is $\frac{45}{60}$.
### Examples
This type of probability calculation is useful in many real-world scenarios, such as planning events or managing resources. For example, a party planner might use this to determine the likelihood that a guest will want either a vegetarian or gluten-free meal, ensuring they have enough options available. Similarly, a store manager could use it to predict how many customers will be interested in a product that is both on sale and eco-friendly, helping them optimize their marketing strategy.
- Use inclusion-exclusion principle: $P(A \cup B) = P(A) + P(B) - P(A \cap B)$.
- Calculate probabilities: $P(A) = \frac{38}{60}$, $P(B) = \frac{13}{60}$, $P(A \cap B) = \frac{6}{60}$.
- Find the final probability: $P(A \cup B) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60} = \boxed{\frac{45}{60}}$.
### Explanation
1. Understanding the Problem
Let's break down this probability problem step by step! We need to find the probability that a customer will be seated at a round table or a table by the window. To do this, we'll use the principle of inclusion-exclusion.
2. Defining Events
Let A be the event that a table is round, and B be the event that a table is by the window. We want to find $P(A \cup B)$, which means the probability of A or B happening.
3. Applying the Inclusion-Exclusion Principle
The formula for $P(A \cup B)$ is: $$P(A \cup B) = P(A) + P(B) - P(A \cap B)$$ This formula helps us avoid counting the tables that are both round and by the window twice.
4. Calculating Individual Probabilities
Now, let's calculate each probability:
$P(A)$ (probability of a round table) = $\frac{\text{Number of round tables}}{\text{Total number of tables}} = \frac{38}{60}$
$P(B)$ (probability of a table by the window) = $\frac{\text{Number of tables by the window}}{\text{Total number of tables}} = \frac{13}{60}$
$P(A \cap B)$ (probability of a round table by the window) = $\frac{\text{Number of round tables by the window}}{\text{Total number of tables}} = \frac{6}{60}$
5. Combining the Probabilities
Plug these values into the formula: $$P(A \cup B) = \frac{38}{60} + \frac{13}{60} - \frac{6}{60}$$
6. Simplifying the Result
Now, let's simplify: $$P(A \cup B) = \frac{38 + 13 - 6}{60} = \frac{45}{60}$$
7. Final Answer
So, the probability that a customer will be seated at a round table or by the window is $\frac{45}{60}$.
### Examples
This type of probability calculation is useful in many real-world scenarios, such as planning events or managing resources. For example, a party planner might use this to determine the likelihood that a guest will want either a vegetarian or gluten-free meal, ensuring they have enough options available. Similarly, a store manager could use it to predict how many customers will be interested in a product that is both on sale and eco-friendly, helping them optimize their marketing strategy.
