Answer :
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].
