Questions: The state medical school has discovered a new test for tuberculosis. (If the test indicates a person has tuberculosis, the test is positive.) Experimentation has shown that the probability of a positive test is 0.78, given that a person has tuberculosis. The probability is 0.07 that the test registers positive, given that the person does not have tuberculosis. Assume that in the general population, the probability that a person has tuberculosis is 0.03. What is the probability that a person chosen at random will fall in the following categories. (a) have tuberculosis and have a positive test (b) not have tuberculosis (c) not have tuberculosis and have a positive test

The state medical school has discovered a new test for tuberculosis. (If the test indicates a person has tuberculosis, the test is positive.) Experimentation has shown that the probability of a positive test is 0.78, given that a person has tuberculosis. The probability is 0.07 that the test registers positive, given that the person does not have tuberculosis. Assume that in the general population, the probability that a person has tuberculosis is 0.03. What is the probability that a person chosen at random will fall in the following categories.
(a) have tuberculosis and have a positive test
(b) not have tuberculosis
(c) not have tuberculosis and have a positive test

Transcript text: The state medical school has discovered a new test for tuberculosis. (If the test indicates a person has tuberculosis, the test is positive.) Experimentation has shown that the probability of a positive test is 0.78 , given that a person has tuberculosis. The probability is 0.07 that the test registers positive, given that the person does not have tuberculosis. Assume that in the general population, the probability that a person has tuberculosis is 0.03 . What is the probability that a person chosen at random will fall in the following categories. (a) have tuberculosis and have a positive test $\square$ (b) not have tuberculosis $\square$ (C) not have tuberculosis and have a positive test $\square$

Solution

Solution Steps

To solve this problem, we will use the given probabilities to calculate the required probabilities for each category using basic probability rules.

(a) To find the probability that a person has tuberculosis and a positive test, we use the formula for the joint probability of two events: P(A and B) = P(A) * P(B|A), where A is having tuberculosis and B is having a positive test.

(b) The probability that a person does not have tuberculosis is simply the complement of the probability that a person has tuberculosis.

(c) To find the probability that a person does not have tuberculosis and has a positive test, we use the formula for the joint probability of two events: P(A' and B) = P(A') * P(B|A'), where A' is not having tuberculosis and B is having a positive test.

Step 1: Probability of Having Tuberculosis and a Positive Test

To find the probability that a person has tuberculosis and has a positive test, we use the formula: \[ P(TB \cap Pos) = P(TB) \cdot P(Pos | TB) \] Substituting the values: \[ P(TB \cap Pos) = 0.03 \cdot 0.78 = 0.0234 \]

Step 2: Probability of Not Having Tuberculosis

The probability that a person does not have tuberculosis is given by: \[ P(\neg TB) = 1 - P(TB) \] Calculating this gives: \[ P(\neg TB) = 1 - 0.03 = 0.97 \]

Step 3: Probability of Not Having Tuberculosis and a Positive Test

To find the probability that a person does not have tuberculosis and has a positive test, we use the formula: \[ P(\neg TB \cap Pos) = P(\neg TB) \cdot P(Pos | \neg TB) \] Substituting the values: \[ P(\neg TB \cap Pos) = 0.97 \cdot 0.07 = 0.0679 \]