Solve your doubts and expand your knowledge with IDNLearn.com's extensive Q&A database. Discover in-depth and reliable answers to all your questions from our knowledgeable community members who are always ready to assist.
Sagot :
To determine which statement is true based on the data provided in the table, we will assess the independence or dependence of events by examining their probabilities and conditional probabilities.
Given:
- Total number of cows: 200
- Cows with seaweed: 124
- Cows without seaweed: 76
- Cows from Farm A: 86
- Cows from Farm B: 114
Step-by-step solution:
1. Calculate the probabilities for Farm A and Farm B:
- [tex]\( P(\text{Farm A}) = \frac{86}{200} = 0.43 \)[/tex]
- [tex]\( P(\text{Farm B}) = \frac{114}{200} = 0.57 \)[/tex]
2. Calculate the probabilities for having feed with and without seaweed:
- [tex]\( P(\text{With Seaweed}) = \frac{124}{200} = 0.62 \)[/tex]
- [tex]\( P(\text{Without Seaweed}) = \frac{76}{200} = 0.38 \)[/tex]
3. Calculate the joint probabilities:
- [tex]\( P(\text{Farm A and With Seaweed}) = \frac{50}{200} = 0.25 \)[/tex]
- [tex]\( P(\text{Farm B and Without Seaweed}) = \frac{40}{200} = 0.2 \)[/tex]
4. Calculate the conditional probabilities:
- [tex]\( P(\text{With Seaweed | Farm A}) = \frac{50}{86} \approx 0.5814 \)[/tex]
- [tex]\( P(\text{Without Seaweed | Farm B}) = \frac{40}{114} \approx 0.3509 \)[/tex]
5. Check for independence:
- For independence of Farm A and having seaweed in its feed:
- Check if [tex]\( P(\text{Farm A and With Seaweed}) = P(\text{Farm A}) \times P(\text{With Seaweed}) \)[/tex]
- [tex]\( P(\text{Farm A}) \times P(\text{With Seaweed}) = 0.43 \times 0.62 = 0.266 \neq 0.25 \)[/tex]
- Thus, Farm A and having seaweed are dependent.
- For independence of having seaweed given Farm A:
- Check if [tex]\( P(\text{With Seaweed | Farm A}) = P(\text{With Seaweed}) \)[/tex]
- [tex]\( P(\text{With Seaweed | Farm A}) = 0.5814 \neq 0.62 \)[/tex]
- Thus, having seaweed and being from Farm A are dependent.
- For independence of Farm B and not having seaweed:
- Check if [tex]\( P(\text{Farm B and Without Seaweed}) = P(\text{Farm B}) \times P(\text{Without Seaweed}) \)[/tex]
- [tex]\( P(\text{Farm B}) \times P(\text{Without Seaweed}) = 0.57 \times 0.38 = 0.2166 \neq 0.2 \)[/tex]
- Thus, Farm B and not having seaweed are dependent.
- For independence of not having seaweed given Farm B:
- Check if [tex]\( P(\text{Without Seaweed | Farm B}) = P(\text{Without Seaweed}) \)[/tex]
- [tex]\( P(\text{Without Seaweed | Farm B}) = 0.3509 \neq 0.38 \)[/tex]
- Thus, not having seaweed and being from Farm B are dependent.
Answer Analysis:
- Statement A: A cow being from Farm A and having seaweed in its feed are dependent because [tex]\( P(\text{Farm A and With Seaweed}) \neq P(\text{Farm A}) \times P(\text{With Seaweed}) \)[/tex]. True
- Statement B: A cow having seaweed in its feed and being from Farm A are independent because [tex]\( P(\text{With Seaweed | Farm A}) = P(\text{With Seaweed}) \)[/tex]. False
- Statement C: A cow being from Farm B and not having seaweed in its feed are dependent because [tex]\( P(\text{Farm B and Without Seaweed}) \neq P(\text{Farm B}) \times P(\text{Without Seaweed}) \)[/tex]. True
- Statement D: A cow not having seaweed in its feed and being from Farm B are independent because [tex]\( P(\text{Without Seaweed | Farm B}) = P(\text{Without Seaweed}) \)[/tex]. False
Therefore, the correct statements are A and C. However, since we need to choose one statement, we select:
A. A cow being from Farm A and having seaweed in its feed are dependent because [tex]\( P(\text{farm A | with seaweed}) \neq P(\text{farm A}) \)[/tex].
Given:
- Total number of cows: 200
- Cows with seaweed: 124
- Cows without seaweed: 76
- Cows from Farm A: 86
- Cows from Farm B: 114
Step-by-step solution:
1. Calculate the probabilities for Farm A and Farm B:
- [tex]\( P(\text{Farm A}) = \frac{86}{200} = 0.43 \)[/tex]
- [tex]\( P(\text{Farm B}) = \frac{114}{200} = 0.57 \)[/tex]
2. Calculate the probabilities for having feed with and without seaweed:
- [tex]\( P(\text{With Seaweed}) = \frac{124}{200} = 0.62 \)[/tex]
- [tex]\( P(\text{Without Seaweed}) = \frac{76}{200} = 0.38 \)[/tex]
3. Calculate the joint probabilities:
- [tex]\( P(\text{Farm A and With Seaweed}) = \frac{50}{200} = 0.25 \)[/tex]
- [tex]\( P(\text{Farm B and Without Seaweed}) = \frac{40}{200} = 0.2 \)[/tex]
4. Calculate the conditional probabilities:
- [tex]\( P(\text{With Seaweed | Farm A}) = \frac{50}{86} \approx 0.5814 \)[/tex]
- [tex]\( P(\text{Without Seaweed | Farm B}) = \frac{40}{114} \approx 0.3509 \)[/tex]
5. Check for independence:
- For independence of Farm A and having seaweed in its feed:
- Check if [tex]\( P(\text{Farm A and With Seaweed}) = P(\text{Farm A}) \times P(\text{With Seaweed}) \)[/tex]
- [tex]\( P(\text{Farm A}) \times P(\text{With Seaweed}) = 0.43 \times 0.62 = 0.266 \neq 0.25 \)[/tex]
- Thus, Farm A and having seaweed are dependent.
- For independence of having seaweed given Farm A:
- Check if [tex]\( P(\text{With Seaweed | Farm A}) = P(\text{With Seaweed}) \)[/tex]
- [tex]\( P(\text{With Seaweed | Farm A}) = 0.5814 \neq 0.62 \)[/tex]
- Thus, having seaweed and being from Farm A are dependent.
- For independence of Farm B and not having seaweed:
- Check if [tex]\( P(\text{Farm B and Without Seaweed}) = P(\text{Farm B}) \times P(\text{Without Seaweed}) \)[/tex]
- [tex]\( P(\text{Farm B}) \times P(\text{Without Seaweed}) = 0.57 \times 0.38 = 0.2166 \neq 0.2 \)[/tex]
- Thus, Farm B and not having seaweed are dependent.
- For independence of not having seaweed given Farm B:
- Check if [tex]\( P(\text{Without Seaweed | Farm B}) = P(\text{Without Seaweed}) \)[/tex]
- [tex]\( P(\text{Without Seaweed | Farm B}) = 0.3509 \neq 0.38 \)[/tex]
- Thus, not having seaweed and being from Farm B are dependent.
Answer Analysis:
- Statement A: A cow being from Farm A and having seaweed in its feed are dependent because [tex]\( P(\text{Farm A and With Seaweed}) \neq P(\text{Farm A}) \times P(\text{With Seaweed}) \)[/tex]. True
- Statement B: A cow having seaweed in its feed and being from Farm A are independent because [tex]\( P(\text{With Seaweed | Farm A}) = P(\text{With Seaweed}) \)[/tex]. False
- Statement C: A cow being from Farm B and not having seaweed in its feed are dependent because [tex]\( P(\text{Farm B and Without Seaweed}) \neq P(\text{Farm B}) \times P(\text{Without Seaweed}) \)[/tex]. True
- Statement D: A cow not having seaweed in its feed and being from Farm B are independent because [tex]\( P(\text{Without Seaweed | Farm B}) = P(\text{Without Seaweed}) \)[/tex]. False
Therefore, the correct statements are A and C. However, since we need to choose one statement, we select:
A. A cow being from Farm A and having seaweed in its feed are dependent because [tex]\( P(\text{farm A | with seaweed}) \neq P(\text{farm A}) \)[/tex].
Thank you for being part of this discussion. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Discover the answers you need at IDNLearn.com. Thanks for visiting, and come back soon for more valuable insights.
