To solve this problem, we can use Bayes' theorem, which helps in finding the probability of an event given prior knowledge of conditions related to the event. Here, we want to find the probability that a randomly selected resident of the Gold dorm is a senior.

Step 1: Define the events and given information

1. Let $S$ represent the event that a student is a senior.
2. Let $J$ represent the event that a student is a junior.
3. Let $U$ represent the event that a student is an underclassman (freshman or sophomore).
4. Let $G$ represent the event that a student lives in the Gold dorm.

The given information can be broken down as follows:

• 10% of the students who choose to live on campus are seniors. So, $P(S) = 0.10$.
• 20% of the students are juniors. So, $P(J) = 0.20$.
• The remaining students are underclassmen (freshmen and sophomores), so $P(U) = 1 - P(S) - P(J) = 1 - 0.10 - 0.20 = 0.70$.

For the probabilities of living in the Gold dorm based on classification:

• 60% of the seniors live in the Gold dorm. So, $P(G | S) = 0.60$.
• 15% of the juniors live in the Gold dorm. So, $P(G | J) = 0.15$.
• 5% of the underclassmen live in the Gold dorm. So, $P(G | U) = 0.05$.

Step 2: Calculate the total probability of a student living in the Gold dorm, $P(G)$

We can use the law of total probability to find $P(G)$: $P(G) = P(G | S) \cdot P(S) + P(G | J) \cdot P(J) + P(G | U) \cdot P(U)$ Substitute the known values: $P(G) = (0.60)(0.10) + (0.15)(0.20) + (0.05)(0.70)$ Calculate each term:

1. $(0.60)(0.10) = 0.06$
2. $(0.15)(0.20) = 0.03$
3. $(0.05)(0.70) = 0.035$

Now add these up: $P(G) = 0.06 + 0.03 + 0.035 = 0.125$

Step 3: Calculate the probability that a student is a senior given that they live in the Gold dorm, $P(S | G)$

We use Bayes' theorem here: $P(S | G) = \frac{P(G | S) \cdot P(S)}{P(G)}$ Substitute the values we have: $P(S | G) = \frac{(0.60)(0.10)}{0.125} = \frac{0.06}{0.125} = 0.48$

Final Answer

The probability that a randomly selected resident of the Gold dorm is a senior is 0.48, or 48%.

In conclusion: If a student lives in the Gold dorm, there is a 48% chance that they are a senior.

Would you like more details on any part of the solution?

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1. How would the probability change if more seniors opted to live in the Gold dorm?
2. What is the probability that a resident of the Gold dorm is a junior?
3. How would Bayes' theorem apply if the probabilities for each group were different?
4. If 50% of freshmen chose the Gold dorm, how would it affect $P(S | G)$?
5. How does the law of total probability help in calculating combined probabilities?

Tip: Understanding conditional probability is key to problems where events depend on prior classifications.