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Let's go through each part of the problem carefully and solve them:

### Problem Overview
Two individuals, Jim and Billy, are attempting to hit a pheasant. The probability of Billy hitting the pheasant is \( P(B) = 0.9 \), while the probability of Jim hitting the pheasant is \( P(J) = 0.6 \). Their events are independent, which means the outcome of one does not affect the other.

#### Part (a)
**Find the probability of both Jim and Billy hitting the pheasant.**

Since the events are independent, the probability of both Jim and Billy hitting the pheasant is the product of their individual probabilities:
\[
P(\text{Both hit}) = P(J) \times P(B) = 0.6 \times 0.9 = 0.54
\]
So, the probability that both Jim and Billy hit the pheasant is **0.5400** (rounded to four decimal places).

#### Part (b)
**Find the probability that both Jim and Billy miss the pheasant.**

The probability that Jim misses the pheasant is \( P(J^c) = 1 - P(J) = 1 - 0.6 = 0.4 \), and the probability that Billy misses is \( P(B^c) = 1 - P(B) = 1 - 0.9 = 0.1 \). Again, since the events are independent, the probability that both miss is:
\[
P(\text{Both miss}) = P(J^c) \times P(B^c) = 0.4 \times 0.1 = 0.04
\]
So, the probability that both miss is **0.0400** (rounded to four decimal places).

#### Part (c)
**Find the probability that the pheasant is hit.**

The pheasant is hit if at least one of Jim or Billy hits it. To find this probability, we can calculate the complement of the probability that both miss. This is given by:
\[
P(\text{At least one hits}) = 1 - P(\text{Both miss}) = 1 - 0.04 = 0.96
\]
So, the probability that the pheasant is hit is **0.9600** (rounded to four decimal places).

### Summary of Answers:
- **Part (a):** 0.5400
- **Part (b):** 0.0400
- **Part (c):** 0.9600

Would you like a detailed breakdown of any of these parts, or do you have any other questions?

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Here are some related questions for further exploration:
1. What is the probability that only Jim hits the pheasant?
2. What is the probability that only Billy hits the pheasant?
3. How would the probabilities change if the events were not independent?
4. What is the probability that exactly one of Jim or Billy hits the pheasant?
5. How could these probabilities be represented using a probability tree diagram?

**Tip:** When working with independent events, the product of individual probabilities gives the probability of both events happening together.

Two buddies, Jim and Billy, embark on a pheasant-hunting trip. Billy has a 0.9 probability of hitting the pheasant, and Jim has a 0.6 probability. The events are independent. Find the probability that both hit the pheasant, both miss, and the probability that the pheasant is hit.

Calculate the probability of both Jim and Billy hitting or missing a pheasant, and the probability that at least one hits it. This problem demonstrates the use of probability rules for independent events with real-world examples.

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