Question

A clinic specializes in testing for a Disease D. The result of the test can be either positive (+ve) or negative (-ve). A study reveals that if a person suffers from the Disease D, the test result in that clinic comes up +ve 80% of the time and negative 20% of the time. If a person is not suffering from the Disease D, the test comes out positive 10% of the time \& negative 90% of the time. It is also known among the general population the disease D occurs in 30% of the individuals. If the person tests positive for D in that clinic, the probability that he/she actually suffers from the Disease D is __________ (Round off to 2 decimal places).

Hint:

Bayes' Theorem questions are common in probability. The key is to correctly identify the prior probability (P(D)), the conditional probabilities (P(T|D) and P(T|D')), and then calculate the total probability of the evidence (P(T)) before finding the final posterior probability (P(D|T)).

Answer 1

Correct Answer: 0.7

Step 1: Understanding the Question:
This is a conditional probability problem that can be solved using Bayes' Theorem. We are given the prevalence of a disease and the accuracy of a test (its true positive and false positive rates). We need to find the probability that a person actually has the disease given that they tested positive.
Step 2: Key Formula or Approach:
Let D be the event that a person has the Disease D.
Let T be the event that the test result is positive.
We want to find P(D | T), the probability of having the disease given a positive test.
Bayes' Theorem is given by: \[ P(D | T) = \frac{P(T | D) \times P(D)}{P(T)} \] To find P(T), we use the Law of Total Probability: \[ P(T) = P(T | D)P(D) + P(T | D')P(D') \] where D' is the event that a person does not have the disease.
Step 3: Detailed Explanation:
First, let's list the probabilities given in the problem:
- Probability of having the disease, P(D) = 0.30.
- Probability of not having the disease, P(D') = 1 - P(D) = 1 - 0.30 = 0.70.
- Probability of testing positive given the person has the disease (True Positive Rate), P(T | D) = 0.80.
- Probability of testing positive given the person does not have the disease (False Positive Rate), P(T | D') = 0.10.
Next, we calculate the overall probability of testing positive, P(T): \[ P(T) = P(T | D)P(D) + P(T | D')P(D') \] \[ P(T) = (0.80 \times 0.30) + (0.10 \times 0.70) \] \[ P(T) = 0.24 + 0.07 = 0.31 \] Now, we can apply Bayes' Theorem to find P(D | T): \[ P(D | T) = \frac{P(T | D) \times P(D)}{P(T)} \] \[ P(D | T) = \frac{0.80 \times 0.30}{0.31} \] \[ P(D | T) = \frac{0.24}{0.31} \] \[ P(D | T) \approx 0.77419... \] Step 4: Final Answer:
Rounding the result to 2 decimal places, we get 0.77.