Suppose that at UVA, 74% of all undergraduates are in the College, 11% are in Engineering, 7% are in Commerce, 3% are in Nursing, and 5% are in Architecture. In each school, the percentage of females is as follows: 56% in the College, 25% in Engineering, 48% in Commerce, 85% in Nursing, and 30% in Architecture. If a randomly selected student is male, what is the probability that he's from the College?

We are given the distribution of students across different schools at UVA and the percentage of females in each school. The task is to find the probability that a randomly selected male student is from the College.

### Step 1: Define the problem
We need to find \( P(\text{College} \mid \text{Male}) \), i.e., the probability that a student is in the College given that the student is male.

Using Bayes' Theorem, we can express this as:

\[
P(\text{College} \mid \text{Male}) = \frac{P(\text{Male} \mid \text{College}) \cdot P(\text{College})}{P(\text{Male})}
\]

Where:
- \( P(\text{Male} \mid \text{College}) \) is the probability that a student is male given they are in the College.
- \( P(\text{College}) \) is the prior probability that a student is in the College.
- \( P(\text{Male}) \) is the total probability that a randomly selected student is male.

### Step 2: Calculate \( P(\text{Male} \mid \text{College}) \)
We are told that 56% of students in the College are female, so the probability that a student in the College is male is:

\[
P(\text{Male} \mid \text{College}) = 1 - 0.56 = 0.44
\]

### Step 3: Calculate \( P(\text{Male}) \) (total probability of selecting a male)
To find \( P(\text{Male}) \), we need to use the law of total probability. The probability that a randomly selected student is male is the sum of the probabilities that the student is male from each school, weighted by the percentage of students in each school:

\[
P(\text{Male}) = P(\text{Male} \mid \text{College}) \cdot P(\text{College}) + P(\text{Male} \mid \text{Engineering}) \cdot P(\text{Engineering}) + \ldots
\]

We know the percentages of males in each school:
- College: \( 1 - 0.56 = 0.44 \)
- Engineering: \( 1 - 0.25 = 0.75 \)
- Commerce: \( 1 - 0.48 = 0.52 \)
- Nursing: \( 1 - 0.85 = 0.15 \)
- Architecture: \( 1 - 0.30 = 0.70 \)

Now we calculate \( P(\text{Male}) \):

\[
P(\text{Male}) = (0.44 \times 0.74) + (0.75 \times 0.11) + (0.52 \times 0.07) + (0.15 \times 0.03) + (0.70 \times 0.05)
\]

### Step 4: Perform the calculation

\[
P(\text{Male}) = (0.44 \times 0.74) + (0.75 \times 0.11) + (0.52 \times 0.07) + (0.15 \times 0.03) + (0.70 \times 0.05)
\]
\[
P(\text{Male}) = 0.3256 + 0.0825 + 0.0364 + 0.0045 + 0.035
\]
\[
P(\text{Male}) = 0.484
\]

### Step 5: Calculate the posterior probability
Now that we have \( P(\text{Male}) = 0.484 \), we can calculate \( P(\text{College} \mid \text{Male}) \):

\[
P(\text{College} \mid \text{Male}) = \frac{P(\text{Male} \mid \text{College}) \cdot P(\text{College})}{P(\text{Male})} = \frac{0.44 \times 0.74}{0.484}
\]

\[
P(\text{College} \mid \text{Male}) = \frac{0.3256}{0.484} \approx 0.6728
\]

### Final Answer:
The probability that a randomly selected male student is from the College is approximately \( 0.673 \), or 67.3%.

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Would you like further details or have any questions on this? Here are five related questions you might find useful:

1. What is Bayes' Theorem and how is it applied in probability?
2. How do you calculate the total probability in a weighted system?
3. What is the difference between conditional and marginal probability?
4. How does the law of total probability work in different applications?
5. What are other real-world applications of conditional probability?

**Tip:** Always ensure to check if the sum of probabilities (like \( P(\text{Male}) \)) makes sense as a total across all scenarios.

Solve a conditional probability problem using Bayes' Theorem. Find the probability that a male student at UVA is from the College, given distribution data across different schools. This problem involves calculating total probability and applying Bayes' Theorem.

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