https://file.aiquickdraw.com/mathbot/file/1728554207341u0dt9idr.jpg

Let's analyze the problem step by step.

The scenario involves two people, Jim and Billy, who are shooting at a pheasant. We know the following probabilities:

- The probability of **Billy** hitting the pheasant: \( P(B) = 0.9 \)
- The probability of **Jim** hitting the pheasant: \( P(J) = 0.6 \)

These events are independent.

### Part (a): Probability of both Jim and Billy hitting the pheasant
Since the events are independent, the probability of both hitting the pheasant is the product of their individual probabilities:
\[
P(\text{Jim and Billy both hit}) = P(J) \times P(B) = 0.6 \times 0.9 = 0.54
\]
So, the probability of both hitting the pheasant is **0.5400**.

### Part (b): Probability that both Jim and Billy miss the pheasant
The probability that they both **miss** is the product of the probabilities that each misses. The probability that Jim misses is \(1 - P(J) = 0.4\), and the probability that Billy misses is \(1 - P(B) = 0.1\). So:
\[
P(\text{Jim and Billy both miss}) = (1 - P(J)) \times (1 - P(B)) = 0.4 \times 0.1 = 0.04
\]
Thus, the probability that both miss the pheasant is **0.0400**.

### Part (c): Probability that the pheasant is hit
To find the probability that the pheasant is hit, we can calculate the complement of the event that both Jim and Billy miss. That is, the pheasant is hit if at least one of them hits it. The probability that both miss is \(0.04\), so the probability that the pheasant is hit is:
\[
P(\text{Pheasant is hit}) = 1 - P(\text{Both miss}) = 1 - 0.04 = 0.96
\]
So, the probability that the pheasant is hit is **0.9600**.

### Summary:
- Part (a): Probability that both Jim and Billy hit = **0.5400**
- Part (b): Probability that both Jim and Billy miss = **0.0400**
- Part (c): Probability that the pheasant is hit = **0.9600**

Let me know if you need further details or have any questions! Here are some related questions to explore further:

1. What if Jim's shooting probability improves to 0.8? How would that change the probabilities?
2. What is the probability that only Billy hits the pheasant?
3. What is the probability that only Jim hits the pheasant?
4. How would the scenario change if the events were not independent?
5. Can we calculate the expected number of hits if both shoot multiple times?

**Tip**: When dealing with independent events, always use the product rule for probabilities of simultaneous outcomes.

Two buddies, Jim and Billy, embark on a pheasant-hunting trip. As they walk through the tall grass of the fescue prairie, a large male ring-necked pheasant flies up from his roost. Both Jim and Billy raise their shotguns, simultaneously aim, then shoot at the pheasant. Billy is a better shot than Jim. Suppose the probability of Billy hitting the pheasant is 0.9, and the probability of Jim hitting the pheasant is 0.6. Assume the events 'Billy shooting the pheasant' and 'Jim shooting the pheasant' are independent events.

Part (a) Find the probability of both Jim and Billy hitting the pheasant.
Part (b) What is the probability that both Jim and Billy miss the pheasant?
Part (c) What is the probability that the pheasant is hit? Think about all the scenarios that result in the pheasant being hit.

Learn how to calculate the probability in a pheasant-hunting scenario where two shooters, Jim and Billy, have independent chances of hitting the pheasant. This problem involves probability multiplication for independent events and determining complementary events.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation