Problem 91 - 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 probability is 0.9, which matches option (B).

Step 1: Understand the Problem

We need to find the probability that a person actually has the disease given that both tests are positive. This can be calculated using Bayes' theorem.

Step 2: Define the Known Probabilities

Let event H = person has the disease, and event T = both tests are positive. - P(H) = 0.1 (probability of having the disease) - P(T|H) = 0.9 * 0.9 = 0.81 (both tests are correct if the person has the disease) - P(T|H') = 0.1 * 0.1 = 0.01 (both tests are correct if the person does not have the disease)

Step 3: Calculate Total Probability of Positive Tests P(T)

Use the law of total probability:\[ P(T) = P(T|H) \times P(H) + P(T|H') \times P(H') \]\[ P(T) = 0.81 \times 0.1 + 0.01 \times 0.9 = 0.081 + 0.009 = 0.09 \].

Step 4: Apply Bayes' Theorem

Bayes' theorem states:\[ P(H|T) = \frac{P(T|H) \times P(H)}{P(T)} \]Substitute the values:\[ P(H|T) = \frac{0.81 \times 0.1}{0.09} = \frac{0.081}{0.09} = 0.9 \].

Step 5: Conclusion

The probability that the person really has the illness given both tests are positive is 0.9, which matches option (B) in the choices provided.

Key Concepts

Understanding Probability

Probability is the measure of the likelihood that an event will occur. It ranges from 0 to 1, where 0 indicates impossibility, and 1 indicates certainty. In simpler terms, it tells you how likely something is to happen.
Imagine you have a bag with 3 red and 2 green marbles. If you randomly pick one, the chances of picking a red marble are the probability of that event. Here, it would be the number of red marbles divided by the total number of marbles, or 3/5. This means you have a 60% chance of picking a red marble.
Probability helps in understanding and predicting outcomes in uncertain situations by providing a numerical value to expectations. In real-life situations, like medical tests, probability plays a crucial role in interpreting test results and diagnosing patients accurately.

Decoding Conditional Probability

Conditional probability is the chance of an event occurring given that another event has already occurred. The key here is the dependency of one event on the outcome of another. This is formally expressed as \( P(A|B) \), which denotes the probability of event A happening given that event B has occurred.
Regarding our original exercise, conditional probability aids in understanding how probable it is that a person actually has a disease when both tests administered show positive results. Here, both tests' outcomes influence the final probability of having the disease. For instance, testing positive on one test alone might not suffice, but both tests being positive significantly increases the reliability of the diagnosis due to the concept of conditional probability. This is exactly what Bayes' Theorem addresses by combining both test results to give a single, reliable probability value. This insight can be immensely valuable in fields such as healthcare, finance, and any domain where outcomes depend on prior conditions.

Exploring the Law of Total Probability

The Law of Total Probability is essential when evaluating the overall likelihood of an outcome derived from various possible events. It breaks down complex probability problems into simpler parts by considering all possible ways an event can occur.
In our exercise, the use of the law of total probability allows us to calculate the overall probability of both tests turning out positive, regardless of whether the person actually has the disease. This total probability is crucial for applying Bayes' Theorem later on. Mathematically, it's depicted as:

• The probability of positive tests if the person has the disease, multiplied by the probability of having the disease.
• The probability of positive tests if the person does not have the disease, multiplied by the probability of not having the disease.

Summing these probabilities gives us a comprehensive view. The law thus helps us weigh all possible scenarios and their contributions to the ultimate outcome. Understanding this concept is vital for solving complex problems where multiple scenarios or conditions affect the final probability.