Problem 154 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The probability that \(A\) speaks truth is \(\frac{4}{5}\), while this probability for \(B\) is \(\frac{3}{4}\). The probability that they contradict each other when asked to speak on a fact is \([\mathbf{2 0 0 4}]\) (A) \(\frac{3}{20}\) (B) \(\frac{1}{5}\) (C) \(\frac{7}{20}\) (D) \(\frac{4}{5}\)

Step-by-Step Solution

Answer

The probability is \(\frac{7}{20}\). (Option C)

Step 1: Identify Probabilities

First, let's denote the probability that \(A\) speaks the truth as \(P(T_A) = \frac{4}{5}\), which means the probability that \(A\) lies, \(P(F_A)\), is \(1 - \frac{4}{5} = \frac{1}{5}\). Similarly, for \(B\), the probability that \(B\) speaks the truth is \(P(T_B) = \frac{3}{4}\), thus the probability that \(B\) lies, \(P(F_B)\), is \(1 - \frac{3}{4} = \frac{1}{4}\).

Step 2: Define Contradiction Scenarios

For \(A\) and \(B\) to contradict each other, one must be telling the truth while the other lies. Thus, there are two possible scenarios: \(A\) tells the truth and \(B\) lies, or \(A\) lies and \(B\) tells the truth.

Step 3: Calculate Probability for Each Scenario

Calculate the probability of scenario 1 where \(A\) tells the truth and \(B\) lies: \[ P(T_A \cap F_B) = P(T_A) \cdot P(F_B) = \frac{4}{5} \cdot \frac{1}{4} = \frac{4}{20}. \]Then, calculate the probability of scenario 2 where \(A\) lies and \(B\) tells the truth: \[ P(F_A \cap T_B) = P(F_A) \cdot P(T_B) = \frac{1}{5} \cdot \frac{3}{4} = \frac{3}{20}. \]

Step 4: Calculate Total Probability of Contradiction

Add the probabilities of the two scenarios from Step 3 to get the total probability that \(A\) and \(B\) contradict each other:\[ P(\text{contradiction}) = P(T_A \cap F_B) + P(F_A \cap T_B) = \frac{4}{20} + \frac{3}{20} = \frac{7}{20}. \]

Key Concepts

Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. This concept is key in problems involving dependent events. In our exercise, knowing that one person tells the truth affects the chances of the other lying to create a contradiction. For instance, if we know that person \( A \) tells the truth, the remaining chance of person \( B \) lying is used. The formula for conditional probability is:

• \( P(B | A) = \frac{P(A \cap B)}{P(A)} \)

This formula can reshape and simplify complex probability scenarios, such as deciding the likelihood of contradictions between statements of two people. This exercise made use of calculating probabilities based on given conditions instead of independent coexistence.

Independent Events

In probability, independent events are those whose occurrence does not influence each other's chances. The exercise implicitly assumes the independence between \( A \)'s and \( B \)'s truthfulness or lying. Under this assumption, the likelihood of them contradicting is straightforwardly computed by multiplying their respective probabilities. This is expressed as:

• \( P(A \cap B) = P(A) \cdot P(B) \)

Where the joint probability is simply the product of individual probabilities. Given that in this exercise, whether \( A \) speaks the truth is independent of whether \( B \) does, it justifies the simple multiplication of probabilities. However, it's crucial to verify independence in practical scenarios. If events were found to be dependent, this approach would not hold true.

Contradictory Statements Analysis

Analyzing contradictory statements involves understanding how probabilities of certain exclusive events affect the likelihood of them occurring together as opposites. In our exercise, \( A \) and \( B \) can speak either truthfully or falsely, offering two scenarios of contradiction. Importantly, for them to contradict each other, one must be lying while the other tells the truth. This situation is characterized by non-overlapping occurrences which:

• \( A \) tells the truth and \( B \) lies
• \( A \) lies and \( B \) tells the truth

By decomposing the problem into these distinct scenarios, we ensure no double-counting occurs and determine the most accurate probability of contradiction. The sum of the individual contradiction probabilities gives us a clear understanding of the total chance that \( A \) and \( B \) will disagree based on their probabilities of stating facts accurately.