A medical test is available to determine whether a patient has a certain disease. To determine the accuracy of the test, a total of 10300 people are tested. Only 200 of these people have the disease, while the other 10100 are disease free. Of the disease-free people, 10000 get a negative result, and 100 get a positive result. The 200 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.
The probability that the test gives the correct result for a person who does not have the disease is [?]

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 consider the outcomes for disease-free individuals.

Out of 10,100 disease-free people:

• 10,000 individuals received a negative test result (correct result).
• 100 individuals received a positive test result (incorrect result).

The probability that the test gives the correct result for a person who does not have the disease is:

\[ P(\text{Correct Result | No Disease}) = \frac{\text{Number of Correct Negative Results}}{\text{Total Number of Disease-Free Individuals}} = \frac{10,000}{10,100} \]

Now, let's calculate this probability:

\[ P(\text{Correct Result | No Disease}) = \frac{10,000}{10,100} \approx 0.9901 \]

Therefore, the probability that the test gives the correct result for a person who does not have the disease is approximately:

\[ \boxed{0.9901} \]

Answer 2

(b)Find the probability that the test gives the correct result for a person who has the disease.
The probability that the test gives the correct result for a person who has the disease is [?]

Answer 3

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

Out of the 200 people with the disease:

• All 200 individuals received a positive test result (correct result).

The probability that the test gives the correct result for a person who has the disease is calculated as follows:

\[ P(\text{Correct Result | Has Disease}) = \frac{\text{Number of Correct Positive Results}}{\text{Total Number of People with Disease}} = \frac{200}{200} \]

Now, let's perform the calculation:

\[ P(\text{Correct Result | Has Disease}) = \frac{200}{200} = 1 \]

Therefore, the probability that the test gives the correct result for a person who has the disease is:

\[ \boxed{1.0000} \]

Answer 4

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

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. We need to calculate:

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

Step 1: Calculate the required probabilities

1. Prior Probability of Having the Disease, \(P(\text{Has Disease})\): \[ P(\text{Has Disease}) = \frac{\text{Number of People with Disease}}{\text{Total Number of People}} = \frac{200}{10300} \approx 0.019417 \]

2. Probability of Testing Positive Given Disease, \(P(\text{Positive Result | Has Disease})\): \[ P(\text{Positive Result | Has Disease}) = 1 \quad (\text{as all 200 with the disease tested positive}) \]

3. Probability of Testing Positive, \(P(\text{Positive Result})\): This can be determined by considering both those who have and don’t have the disease:

   • Positive results from those who have the disease: 200
   • Positive results from those who do not have the disease: 100 (from the 10,100 disease-free individuals)

   Therefore: \[ P(\text{Positive Result}) = \frac{200 + 100}{10300} = \frac{300}{10300} \approx 0.029126 \]

Step 2: Apply Bayes' Theorem

Now we can plug these values into Bayes' theorem:

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

Substituting the known values: \[ P(\text{Has Disease | Positive Result}) = \frac{1 \cdot \frac{200}{10300}}{\frac{300}{10300}} = \frac{200}{300} = \frac{2}{3} \approx 0.6667 \]

Thus, the probability that a person actually has the disease given that they received a positive result is:

\[ \boxed{0.6667} \]