To determine whether eating breakfast and passing the test are independent or dependent events, we can look at the probabilities of passing the test given whether or not a student ate breakfast.
First, we need to calculate the total number of students in each category.
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Total number of students who ate breakfast:
- Passed: 122
- Failed: 10
- Total who ate breakfast = 122 + 10 = 132
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Total number of students who did not eat breakfast:
- Passed: 58
- Failed: 15
- Total who did not eat breakfast = 58 + 15 = 73
Now we can calculate the overall totals:
- Total students:
- Total = 132 + 73 = 205
Next, let's calculate the probability of passing the test for both groups (those who ate breakfast and those who did not):
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Probability of passing given that a student ate breakfast (P(Pass | Eat Breakfast)): \[ P(Pass | Eat Breakfast) = \frac{\text{Number who passed and ate breakfast}}{\text{Total who ate breakfast}} = \frac{122}{132} \]
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Probability of passing given that a student did not eat breakfast (P(Pass | Did Not Eat Breakfast)): \[ P(Pass | Did Not Eat Breakfast) = \frac{\text{Number who passed and did not eat breakfast}}{\text{Total who did not eat breakfast}} = \frac{58}{73} \]
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Total probability of passing (P(Pass)): \[ P(Pass) = \frac{\text{Total students who passed}}{\text{Total students}} = \frac{122 + 58}{205} = \frac{180}{205} \]
To check for independence, we need to see if: \[ P(Pass | Eat Breakfast) = P(Pass) \quad \text{and} \quad P(Pass | Did Not Eat Breakfast) = P(Pass) \]
Calculating the probabilities for passing when eating breakfast and not eating breakfast:
- \( P(Pass | Eat Breakfast) = \frac{122}{132} \approx 0.9242 \)
- \( P(Pass | Did Not Eat Breakfast) = \frac{58}{73} \approx 0.7959 \)
- \( P(Pass) = \frac{180}{205} \approx 0.8780 \)
Since:
- \( P(Pass | Eat Breakfast) \approx 0.9242 \) is not equal to \( P(Pass) \approx 0.8780 \)
- \( P(Pass | Did Not Eat Breakfast) \approx 0.7959 \) is also not equal to \( P(Pass) \approx 0.8780 \)
Conclusion
Because the probabilities differ depending on whether or not the students ate breakfast, we conclude that eating breakfast and passing the test are dependent events.