Suppose that 60% of international students and 50% of local students in a university prefer online format for office hours. A group of 20 students in the university, including 12 international students and 8 local students were asked their preferred format for office hours. A response is randomly picked from the group and is in-person. What is the probability that the response is from a local student?

We can solve this problem using **Bayes' Theorem**, which helps find the probability of an event given that another event has occurred.

Let's define the events:
- \( A \): The event that the student is local.
- \( B \): The event that the response is in-person.

We want to find the conditional probability \( P(A|B) \), the probability that the student is local given that their response is in-person. Using Bayes' Theorem:

\[
P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)}
\]

Where:
- \( P(A) \) is the prior probability that the student is local.
- \( P(B|A) \) is the probability that a local student prefers in-person format.
- \( P(B) \) is the total probability that a randomly chosen student prefers in-person format (whether local or international).

### Step 1: Calculate \( P(A) \)
The group consists of 12 international students and 8 local students, so:
\[
P(A) = \frac{8}{20} = 0.4
\]

### Step 2: Calculate \( P(B|A) \)
We know that 50% of local students prefer online format, so 50% prefer in-person format. Thus:
\[
P(B|A) = 0.5
\]

### Step 3: Calculate \( P(B) \)
We need to find the total probability that a student prefers in-person format, which is the sum of the probabilities for international and local students. Let:
- \( P(\text{Intl}) \) represent the event that a student is international.
- \( P(B|\text{Intl}) \) is the probability that an international student prefers in-person format.

We know:
\[
P(\text{Intl}) = \frac{12}{20} = 0.6
\]
\[
P(B|\text{Intl}) = 1 - 0.6 = 0.4
\]

Now, using the law of total probability:
\[
P(B) = P(B|A) \cdot P(A) + P(B|\text{Intl}) \cdot P(\text{Intl})
\]
\[
P(B) = (0.5 \times 0.4) + (0.4 \times 0.6) = 0.2 + 0.24 = 0.44
\]

### Step 4: Apply Bayes' Theorem
Now we can apply Bayes' Theorem:
\[
P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} = \frac{0.5 \times 0.4}{0.44} = \frac{0.2}{0.44} \approx 0.4545
\]

Thus, the probability that the response is from a local student given that it is in-person is approximately **45.45%**.

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Would you like more details or have any questions? Here are five related questions to explore the topic further:

1. How would the result change if a higher percentage of international students preferred in-person office hours?
2. What if the group of students had a different proportion of local and international students?
3. How does the use of Bayes' Theorem help in decision-making in real-world problems?
4. What if the percentage of local students who prefer online format changed to 60%? How would that impact the result?
5. How would you calculate the probability if you knew the exact number of students who prefer in-person office hours?

**Tip**: Bayes' Theorem is useful in various real-world applications like medical testing, spam filtering, and decision-making under uncertainty.

Solve a probability problem using Bayes' Theorem: Given a group of 12 international and 8 local students, calculate the probability that an in-person response comes from a local student. Understand key concepts like conditional probability, Bayes' Theorem, and the law of total probability. Suitable for high school students in Grades 11-12.

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