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.