Problem 46 - 71 JEE Main Mathematics Online (2020 - 2012) & Offline (2018 - 2002) Chapterwise + Topicwise Solved Papers

Question

The probability that \(A\) speaks truth is \(\frac{4}{5}\), while the probability for \(B\) is \(\frac{3}{4}\). The probability that they contradict each other when asked to speak on a fact is (a) \(\frac{4}{5}\) (b) \(\frac{1}{5}\) (c) \(\frac{7}{20}\) (d) \(\frac{3}{20}\)

Step-by-Step Solution

Answer

The probability is \( \frac{7}{20} \) (option c).

Step 1: Understanding the Problem

We need to find the probability that two individuals, A and B, contradict each other when stating a fact. We know each person's probability of telling the truth. Thus, each person could either tell the truth or lie.

Step 2: Calculating Probability of A Telling a Lie

The probability that A speaks the truth is given as \( \frac{4}{5} \). So, the probability that A lies is the complement: \( 1 - \frac{4}{5} = \frac{1}{5} \).

Step 3: Calculating Probability of B Telling a Lie

Similarly, the probability that B speaks the truth is \( \frac{3}{4} \). Thus, the probability that B lies is \( 1 - \frac{3}{4} = \frac{1}{4} \).

Step 4: Combinations of Contradiction

A contradiction happens in either of these two cases:1. A tells the truth (probability \( \frac{4}{5} \)) and B lies (probability \( \frac{1}{4} \)).2. A lies (probability \( \frac{1}{5} \)) and B tells the truth (probability \( \frac{3}{4} \)).

Step 5: Calculating Total Probability for Contradiction Case 1

Calculate the probability that A tells the truth and B tells a lie: \( \frac{4}{5} \times \frac{1}{4} = \frac{4}{20} = \frac{1}{5} \).

Step 6: Calculating Total Probability for Contradiction Case 2

Calculate the probability that A lies and B tells the truth: \( \frac{1}{5} \times \frac{3}{4} = \frac{3}{20} \).

Step 7: Adding Up Probabilities

The total probability that they contradict each other is the sum of the two cases: \( \frac{1}{5} + \frac{3}{20} = \frac{4}{20} + \frac{3}{20} = \frac{7}{20} \).

Step 8: Conclusion

The probability that A and B contradict each other is \( \frac{7}{20} \). Thus, the answer is option (c): \( \frac{7}{20} \).

Key Concepts

Probability of Contradiction

To understand the probability of contradiction between two people like A and B, we consider situations where they have opposite statements about the same fact. For instance, if one says "Yes" and the other says "No."
A contradiction can occur when:

• Individual A speaks the truth but Individual B tells a lie.
• Individual A tells a lie but Individual B speaks the truth.

In our exercise, we calculate the probability of these events separately and then add them together to find the overall probability. This approach ensures that we cover all scenarios in which a contradiction could occur. Always remember that contradictions arise from opposing statements, which in this context, depend on whether A and B are truthful or not.

Complement Rule

The Complement Rule is a helpful tool in probability that allows us to find the probability of an event not happening, using the probability of it happening. Think of it as flipping a coin; if the probability of heads is known, the probability of tails is simply what's left over from certainty.
In mathematical terms, if the probability of event A happening is given as \( P(A) \), then the probability of event A not happening is \( 1 - P(A) \).
In the provided problem, the probability of A telling a lie is the complement of A telling the truth, calculated as \( 1 - \frac{4}{5} = \frac{1}{5} \). Similarly, the probability of B telling a lie is \( 1 - \frac{3}{4} = \frac{1}{4} \). Using the Complement Rule simplifies finding these complementary probabilities and is crucial in calculating the contradiction probability.

Independent Events

Independent events in probability are events whose outcomes do not affect each other. When calculating probabilities involving two independent events, their individual probabilities are simply multiplied together.
In our exercise, A and B's truth-telling tendencies are independent. That means whether A tells the truth or not does not affect B's truthfulness, and vice versa.

• The probability of A telling the truth and B lying: \( \frac{4}{5} \times \frac{1}{4} \).
• The probability of A lying and B telling the truth: \( \frac{1}{5} \times \frac{3}{4} \).

These computations assume independence and allow adding up these results to get the overall contradiction probability of \( \frac{7}{20} \). Independent events simplify probability calculations substantially by enabling this straightforward multiplication approach.