Harry’s backyard pond contains 6 goldfish and 4 catfish. All fish are equally likely to be caught. (a) Suppose that Harry catches a total of 5 fish (no fish are thrown back). Let G be the event that Harry catches exactly 3 goldfish. What is the probability P(G)? (b) Assume event G occurs in the first catch and suppose that all 5 fish are returned to the pond. Harry starts fishing again. This time he catches a total of 3 fish. Let A be the event that among the caught set of 3 fish, exactly 2 goldfish are included that were also caught in the first catch? (Assume that fish do not learn from experience.) Find P(A).

Let's solve each part of the problem step-by-step.

### (a) Probability that Harry catches exactly 3 goldfish in a total of 5 fish

Given:
- 6 goldfish and 4 catfish in the pond.
- Harry catches a total of 5 fish.
- \( G \) is the event that Harry catches exactly 3 goldfish.

To find \( P(G) \), we need to compute the probability of catching exactly 3 goldfish and 2 catfish out of the total 5 fish.

1. **Number of ways to choose 3 goldfish from 6**:  
   \[
   \binom{6}{3} = \frac{6!}{3!(6-3)!} = \frac{6 \times 5 \times 4}{3 \times 2 \times 1} = 20
   \]

2. **Number of ways to choose 2 catfish from 4**:  
   \[
   \binom{4}{2} = \frac{4!}{2!(4-2)!} = \frac{4 \times 3}{2 \times 1} = 6
   \]

3. **Number of ways to choose any 5 fish from the 10 total fish**:  
   \[
   \binom{10}{5} = \frac{10!}{5!(10-5)!} = \frac{10 \times 9 \times 8 \times 7 \times 6}{5 \times 4 \times 3 \times 2 \times 1} = 252
   \]

4. **Calculating \( P(G) \)**:  
   The probability of catching exactly 3 goldfish and 2 catfish is the ratio of the number of favorable outcomes to the total number of outcomes.

   \[
   P(G) = \frac{\binom{6}{3} \cdot \binom{4}{2}}{\binom{10}{5}}
   = \frac{20 \cdot 6}{252}
   = \frac{120}{252}
   = \frac{10}{21}
   \]

So, \( P(G) = \frac{10}{21} \approx 0.476 \) or **47.6%**.

### (b) Probability that exactly 2 goldfish caught in the second round were also caught in the first round

Given:
- Event \( G \) has occurred, so Harry has already caught 3 goldfish and 2 catfish in the first catch.
- All 5 fish are returned to the pond, and Harry starts fishing again.
- Harry catches a total of 3 fish in the second round.
- \( A \) is the event that among the caught set of 3 fish, exactly 2 goldfish are included that were also caught in the first catch.

To solve this, we need to consider:
- The probability of catching exactly 2 out of the 3 goldfish that were caught in the first round, and 1 goldfish that was not caught in the first round.

1. **Number of ways to choose 2 goldfish from the 3 caught in the first round**:  
   \[
   \binom{3}{2} = \frac{3!}{2!(3-2)!} = \frac{3}{1} = 3
   \]

2. **Number of ways to choose 1 goldfish from the 3 not caught in the first round (since there are 6 total goldfish and 3 were caught in the first round)**:  
   \[
   \binom{3}{1} = 3
   \]

3. **Number of ways to choose any 3 fish from the total 10 fish (the second catch scenario)**:  
   \[
   \binom{10}{3} = \frac{10!}{3!(10-3)!} = \frac{10 \times 9 \times 8}{3 \times 2 \times 1} = 120
   \]

4. **Calculating \( P(A) \)**:  
   The probability of catching exactly 2 goldfish from the first catch and 1 goldfish not caught in the first round is:

   \[
   P(A) = \frac{\binom{3}{2} \cdot \binom{3}{1}}{\binom{10}{3}}
   = \frac{3 \cdot 3}{120}
   = \frac{9}{120}
   = \frac{3}{40}
   \]

So, \( P(A) = \frac{3}{40} = 0.075 \) or **7.5%**.

### Summary of Results

- **(a) \(P(G)\)**: \( \frac{10}{21} \approx 47.6\% \)
- **(b) \(P(A)\)**: \( \frac{3}{40} = 7.5\% \)

Would you like more details or have any questions?

Here are 5 related questions that could expand your understanding:
1. What is the probability that Harry catches 4 goldfish in the first round?
2. How would the probability change if there were more fish in the pond?
3. What is the probability of catching at least one goldfish in the second round?
4. If Harry catches 4 fish instead of 5 in the first round, what would \( P(G) \) be?
5. How would \( P(A) \) change if the pond contained different types of fish?

**Tip**: For problems involving multiple stages, consider each stage separately and use conditional probabilities to combine them.

Explore the probability calculations for catching goldfish and catfish in Harry's pond. Calculate the probability of catching exactly 3 goldfish out of 5 fish and the probability of catching 2 goldfish in a subsequent round. Detailed step-by-step solution included.

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