Let's analyze the information in the table and solve the two problems step-by-step:
Given Table:
| | Positive Result | Negative Result |
|-----------------------------|----------------|-----------------|
| Disease Present | 103 | 15 |
| Disease Not Present | 17 | 207 |
23. Determine the probability that the blood test will detect the disease, if you have the disease.
This question asks for the probability of a positive test result given that the disease is present. We denote this probability as [tex]\( P(\text{Positive} | \text{Disease}) \)[/tex].
1. Identify the number of patients who have the disease and tested positive:
- Disease Present, Positive Result: 103 patients
2. Identify the total number of patients who have the disease:
- Disease Present, Positive Result: 103 patients
- Disease Present, Negative Result: 15 patients
- Total Disease Present = 103 + 15 = 118 patients
3. Calculate the probability:
- [tex]\( P(\text{Positive} | \text{Disease}) = \frac{\text{Number of patients with disease and positive result}}{\text{Total number of patients with disease}} \)[/tex]
- [tex]\( P(\text{Positive} | \text{Disease}) = \frac{103}{118} \approx 0.8729 \)[/tex]
Therefore, the probability that the blood test will detect the disease, if you have the disease, is approximately 0.8729 or 87.29%.
24. What is the probability that you have the disease, if your blood test reports a positive result?
This question asks for the probability of having the disease given a positive test result. We denote this probability as [tex]\( P(\text{Disease} | \text{Positive}) \)[/tex].
1. Identify the number of patients who have a positive test result:
- Disease Present, Positive Result: 103 patients
- Disease Not Present, Positive Result: 17 patients
- Total Positive Results = 103 + 17 = 120 patients
2. Identify the number of patients who have the disease and tested positive:
- Disease Present, Positive Result: 103 patients
3. Calculate the probability:
- [tex]\( P(\text{Disease} | \text{Positive}) = \frac{\text{Number of patients with disease and positive result}}{\text{Total number of positive test results}} \)[/tex]
- [tex]\( P(\text{Disease} | \text{Positive}) = \frac{103}{120} \approx 0.8583 \)[/tex]
Therefore, the probability that you have the disease, if your blood test reports a positive result, is approximately 0.8583 or 85.83%.
To summarize:
1. The probability that the blood test will detect the disease, if you have the disease, is approximately 0.8729 or 87.29%.
2. The probability that you have the disease, if your blood test reports a positive result, is approximately 0.8583 or 85.83%.
