Problem 91 - A Complete Resource Book in Mathematics for JEE Main

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

(B) 0.9

Step 1: Understand the problem

We are tasked with finding the probability that a person actually has the illness given that both tests return positive results. This requires us to use the concept of conditional probability.

Step 2: Define the probabilities

Let the event A be 'the person has the illness' and B be 'both tests are positive.' Given: \( P(A) = 0.1 \). Each test has a \( 90\% \) accuracy, so the probability both tests are positive if you have the disease is \( (0.9)^2 = 0.81 \).

Step 3: Apply Bayes' Theorem

We use Bayes' Theorem: \[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \] where \( P(B|A) = 0.81 \) and \( P(A) = 0.1 \).

Step 4: Find P(B)

To find \( P(B) \), consider both outcomes:1. The person has the disease and both tests are positive: \( P(B|A) \cdot P(A) = 0.81 \cdot 0.1 = 0.081 \).2. The person does not have the disease and tests still show positive. Probability person doesn't have the disease: \( 0.9 \), tests are positive due to errors: \( (0.1)^2 = 0.01 \). Thus, \( P(B|eg A) \cdot P(eg A) = 0.01 \cdot 0.9 = 0.009 \).Add these probabilities: \( P(B) = 0.081 + 0.009 = 0.09 \).

Step 5: Calculate the conditional probability

Substitute back into Bayes' Theorem: \[ P(A|B) = \frac{0.81 \times 0.1}{0.09} = \frac{0.081}{0.09} = 0.9 \].

Step 6: Conclude the Answer

The probability that the person has the illness given both tests are positive is 0.9.

Key Concepts

Conditional Probability

When we talk about conditional probability, we're looking at scenarios where we want to know the likelihood of an event happening, given that another event has already occurred.
Bayes' Theorem is a popular method that helps us find these conditional probabilities. It allows us to update our understanding based on new information.
For example, in our exercise, we use Bayes' Theorem to determine the probability that a person actually has the illness, given that they tested positive twice.

The formula for Bayes' Theorem is: \[ P(A|B) = \frac{P(B|A) \cdot P(A)}{P(B)} \] In this formula:

• \(P(A|B)\) means the probability of event A given event B.
• \(P(B|A)\) is the probability of event B when A is true.
• \(P(A)\) stands for the probability of event A.
• \(P(B)\) is the probability of event B.

In our exercise, event A is having the illness, and event B represents both tests coming back positive. Understanding this theorem helps us see how probabilities change when we know other related information.

Probability Theory

Probability theory is a crucial area of mathematics that deals with analyzing random events. It forms the backbone of understanding how likely events are to occur. It explains everything from simple coin tosses to complex processes like medical testing.

One of the essential concepts of probability theory is the calculation of a probability. When given probabilities, like in our problem, we use mathematical rules to evaluate the outcome based on certain conditions.

Probability is calculated as a fraction between 0 and 1. Zero means an event won't happen, while one guarantees occurrence.
For instance, in the exercise, the probability of having the illness is given as 0.1, or 10%. This is a foundational part that helps us calculate the likelihood of further events, such as the test results.

• Understanding probability theory helps us quantify uncertainty, manage risks, and make informed decisions, all of which are crucial in fields like healthcare, finance, and more.
• In our case, by using the given probabilities and test accuracies, we can assess whether it's likely or unlikely for someone who tested positive twice to have the illness.

Independent Events

In probability, independent events are scenarios where the occurrence of one event doesn’t affect the occurrence of another. This is a significant concept because it simplifies calculations, allowing us to evaluate probabilities separately with more straightforward multiplication rules.

In our exercise, the two tests given to the person are independent events. Each test's result doesn’t influence the outcome of the other. This independence allows us to multiply the probabilities of individual test results to find the overall likelihood of outcomes, such as both tests being positive.

• An independent probability property can often be represented mathematically as: \[ P(A \text{ and } B) = P(A) \cdot P(B) \]
• This formula shows that if two events, A and B, are independent, their joint probability is simply the product of their individual probabilities.
• In our problem, we used the independence to calculate the joint probability of both tests returning positive if the person has the illness, with a 90% accuracy rate per test.

Understanding independent events helps make complex probability problems more manageable, particularly where multiple scenarios or repeated trials are involved.