Problem 46 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

\(10 \%\) of a certain population suffer from a serious disease. A person suspected of the disease is given two independent tests. Each test makes a correct diagnosis \(90 \%\) of the time. The probability that the person really has the illness given that both tests are positive is (A) \(0.5\) (B) \(0.9\) (C) \(0.6\) (D) none of these

Step-by-Step Solution

Answer

The correct answer is (B) 0.9.

Step 1: Define the Probability Notations

Let \( A \) be the event that a person has the illness, and \( B \) be the event that both test results are positive. We are given: \( P(A) = 0.1 \) (probability of having the disease), \( P(B|A) \) (probability of both tests being positive given the disease) is product of two independent tests: \( (0.9) \times (0.9) = 0.81 \), and \( P(B|\text{not } A) \) (both tests positive given no disease) is: \( (0.1) \times (0.1) = 0.01 \).

Step 2: Calculate Probability of Two Positive Tests

Using the Law of Total Probability: \( P(B) = P(B|A) \cdot P(A) + P(B|\text{not } A) \cdot P(\text{not } A) \).Substituting the values we know: \( P(B) = 0.81 \times 0.1 + 0.01 \times 0.9 = 0.081 + 0.009 = 0.09 \).

Step 3: Use Bayes' Theorem to Find Desired Probability

Bayes' Theorem: \( P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \).Plugging in known values: \( P(A|B) = \frac{0.81 \times 0.1}{0.09} = \frac{0.081}{0.09} = 0.9 \).

Step 4: Determine the Correct Answer

The probability that the person really has the illness given that both tests are positive is \(0.9\). Therefore, the correct answer is (B) \(0.9\).

Key Concepts

Probability

Probability is a measure of how likely an event is to occur. It ranges from 0 (the event will not happen) to 1 (the event will certainly happen). In our exercise, the probability of having the disease is 10%, or 0.1 in decimal form. This means that, out of 100 people, roughly 10 may have the disease.

Probability is often represented with the letter "P". For instance, \( P(A) \) refers to the probability of event \( A \) happening. In the solution:

• \( P(A) = 0.1 \) is the probability that a person has the illness.
• \( P(B|A) \) is the probability that both tests are positive given that the person has the illness.
• \( P(B|\text{not } A) \) is the probability that both tests are positive when the person does not have the illness.

Understanding these probabilities helps in applying mathematical procedures, like Bayes' Theorem, to arrive at conclusions about real-world situations.

Independent Events

Events are considered independent when the occurrence of one event does not affect the probability of another. In the exercise, two tests are administered for diagnosing the illness. These tests are described as independent.

The key part here is knowing that each test gives a correct diagnosis 90% of the time, regardless of the other's result. Since the tests are independent, multiply the individual probabilities to find the probability of both tests being positive together:

• \( 0.9 \times 0.9 = 0.81 \).

This is the probability that both tests will correctly show a positive result if the person truly has the disease. This principle of independence is pivotal in calculating \( P(B|A) \).

Law of Total Probability

The Law of Total Probability helps us find the overall probability of an event based on all possible scenarios it can occur. In the exercise, it is used to find \( P(B) \), the probability that both tests are positive, by considering both cases: whether the person has the illness or not.

To use this law, you calculate:

• \( P(B|A) \cdot P(A) \)
• \( P(B|\text{not } A) \cdot P(\text{not } A) \)

Adding these gives the total probability:

• \( P(B) = 0.81 \times 0.1 + 0.01 \times 0.9 = 0.09 \)

This result accounts for both situations—having or not having the disease. Understanding this law helps in breaking down complex probabilities into understandable parts.