A medical test is available to determine whether a patient has a certain disease. To determine the accuracy of the test, a total of 10400 people are tested. Only 100 of these people have the disease, while the other 10300 are disease free. Of the disease-free people, 10000 get a negative result, and 300 get a positive result. The 100 people with the disease all get positive results. Round your answers to four decimal places, if necessary.

Question

(a)Find the probability that the test gives the correct result for a person who does not have the disease.

Answer 1

To find the probability that the test gives the correct result for a person who does not have the disease, we need to look at the results for the disease-free people.

We know:

• Total number of disease-free people: 10,300
• Number of disease-free people who tested negative: 10,000
• Number of disease-free people who tested positive: 300

The "correct result" for a disease-free person is a negative test result. So, the probability that the test gives the correct result for a person who does not have the disease can be calculated using the formula:

\[ P(\text{Correct result} | \text{No disease}) = \frac{\text{Number of disease-free individuals with correct negative results}}{\text{Total number of disease-free individuals}} \]

Plugging in the numbers:

\[ P(\text{Correct result} | \text{No disease}) = \frac{10,000}{10,300} \]

Now performing the calculation:

\[ P(\text{Correct result} | \text{No disease}) = \frac{10,000}{10,300} \approx 0.9709 \]

Thus, the probability that the test gives the correct result for a person who does not have the disease is approximately 0.9709.

Answer 2

(b)Find the probability that the test gives the correct result for a person who has the disease.

Answer 3

To find the probability that the test gives the correct result for a person who has the disease, we again need to consider the test results for those who actually have the disease.

We know:

• Total number of people who have the disease: 100
• Number of people with the disease who tested positive: 100

The "correct result" for a person who has the disease is a positive test result. Therefore, the probability that the test gives the correct result for a person who has the disease can be calculated as follows:

\[ P(\text{Correct result} | \text{Disease}) = \frac{\text{Number of individuals with the disease who tested positive}}{\text{Total number of individuals with the disease}} \]

Substituting the known values:

\[ P(\text{Correct result} | \text{Disease}) = \frac{100}{100} = 1 \]

Thus, the probability that the test gives the correct result for a person who has the disease is 1.0000. This means that all individuals with the disease received a positive test result, which indicates that the test is perfectly accurate for detecting the disease in this sample.

Answer 4

(c)Given that a person gets a positive result, what is the probability that the person actually has the disease?

Answer 5

To find the probability that a person actually has the disease given that they received a positive test result, we can use Bayes' theorem. This requires us to calculate the following:

1. The probability of having the disease (\( P(\text{Disease}) \))
2. The probability of testing positive given that the person has the disease (\( P(\text{Positive} | \text{Disease}) \))
3. The probability of testing positive given that the person does not have the disease (\( P(\text{Positive} | \text{No Disease}) \))
4. The total probability of testing positive (\( P(\text{Positive}) \))

Values we know:

• Total number of people tested: 10,400
• Number with the disease: 100
• Number without the disease: 10,300
• Number of true positives (with disease and tested positive): 100
• Number of false positives (without disease and tested positive): 300

Calculations:

1. Prior probabilities:

   • \( P(\text{Disease}) = \frac{100}{10400} = \frac{1}{104} \)
   • \( P(\text{No Disease}) = \frac{10300}{10400} = \frac{103}{104} \)

2. Sensitivity (true positive rate):

   • \( P(\text{Positive} | \text{Disease}) = \frac{100}{100} = 1 \)

3. False positive rate:

   • \( P(\text{Positive} | \text{No Disease}) = \frac{300}{10300} \)
   • \( P(\text{Positive} | \text{No Disease}) = \frac{300}{10300} \approx 0.0291 \)

4. Total probability of testing positive: \[ P(\text{Positive}) = P(\text{Positive} | \text{Disease}) \cdot P(\text{Disease}) + P(\text{Positive} | \text{No Disease}) \cdot P(\text{No Disease}) \] Substituting the values: \[ P(\text{Positive}) = (1) \cdot \left(\frac{1}{104}\right) + \left(0.0291\right) \cdot \left(\frac{103}{104}\right) \] \[ P(\text{Positive}) = \frac{1}{104} + 0.0291 \cdot \frac{103}{104} \] \[ P(\text{Positive}) = \frac{1}{104} + \frac{0.0291 \cdot 103}{104} \] \[ P(\text{Positive}) \approx \frac{1 + 2.9993}{104} \approx \frac{3.9993}{104} \approx 0.0385 \]

Final Probability using Bayes' Theorem:

Now we can use Bayes' theorem to find \( P(\text{Disease} | \text{Positive}) \):

\[ P(\text{Disease} | \text{Positive}) = \frac{P(\text{Positive} | \text{Disease}) \cdot P(\text{Disease})}{P(\text{Positive})} \]

Substituting in the values we calculated:

\[ P(\text{Disease} | \text{Positive}) = \frac{1 \cdot \frac{1}{104}}{0.0385} \]

Calculating this:

\[ P(\text{Disease} | \text{Positive}) = \frac{\frac{1}{104}}{0.0385} \approx 0.974 \]

Thus, the probability that a person actually has the disease given that they received a positive test result is approximately 0.9740 (or 97.40%).