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

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 [2004] (A) \(\frac{3}{20}\) (B) \(\frac{1}{5}\) (C) \(\frac{7}{20}\) (D) \(\frac{4}{5}\)

Step-by-Step Solution

Answer

The probability of contradiction is \( \frac{7}{20} \), answer (C).

Step 1: Understand the Event

To find the probability that A and B contradict each other, we need to figure out under which circumstances they will say different things.

Step 2: Calculate Probability of A Telling the Truth

The probability that A speaks the truth is \( P(T_A) = \frac{4}{5} \).

Step 3: Calculate Probability of A Not Telling the Truth

If A does not speak the truth, the probability is \( P(F_A) = 1 - P(T_A) = 1 - \frac{4}{5} = \frac{1}{5} \).

Step 4: Calculate Probability of B Telling the Truth

The probability that B speaks the truth is \( P(T_B) = \frac{3}{4} \).

Step 5: Calculate Probability of B Not Telling the Truth

If B does not speak the truth, the probability is \( P(F_B) = 1 - P(T_B) = 1 - \frac{3}{4} = \frac{1}{4} \).

Step 6: Calculate Probability of Contradiction

The events where A and B contradict each other are: A speaks the truth and B does not, or A does not speak the truth and B does. Therefore, the probability is \( P(T_A \wedge F_B) + P(F_A \wedge T_B) = (\frac{4}{5} \times \frac{1}{4}) + (\frac{1}{5} \times \frac{3}{4}) \).

Step 7: Compute Final Probability

Calculate \( (\frac{4}{5} \times \frac{1}{4}) = \frac{4}{20} = \frac{1}{5} \) and \( (\frac{1}{5} \times \frac{3}{4}) = \frac{3}{20} \). Add these probabilities: \( \frac{1}{5} + \frac{3}{20} = \frac{4}{20} + \frac{3}{20} = \frac{7}{20} \).

Key Concepts

Contradiction

Contradiction is when two people say something different about the same fact. In probability, contradictions come into play, especially when dealing with statements or events where differing opinions or outcomes are possible. If two people make statements about the same fact, and those statements do not coincide, they contradict each other.
Each statement has its probability, and a contradiction occurs when one person tells the truth and the other does not. This happens in two scenarios:

• Person A tells the truth and Person B doesn't
• Person B tells the truth and Person A doesn't

By understanding these possibilities, we can use their individual truthfulness probabilities to calculate when they will indeed contradict each other.

Truthfulness

Truthfulness refers to the likelihood or probability that a person is telling the truth. In probability terms, it is the chance that a person's statement about an event or fact is accurate. For any given person, this probability can be calculated if the person's history or behavior is known.
In the given problem, you have:

• Person A's truthfulness as \( \frac{4}{5} \) or 80%.
• Person B's truthfulness as \( \frac{3}{4} \) or 75%.

These percentages translate into how likely they are to speak the truth in any given instance.
However, what they don't tell you directly is the probability of them not telling the truth. This is simply the complement of the truthfulness probability (i.e., \( 1 - \text{truthfulness}\)). By understanding both these probabilities, you can discern their likely behavior and work out cases where they could disagree or contradict.

Conditional Probability

Conditional probability deals with the likelihood of an event occurring, given that another event has already occurred. It’s crucial in scenarios where the probability of one event might depend on the outcome of a previous event.
To illustrate, let's consider how Person A or B's statement affects the probability of contradiction. Suppose:

• "A speaks truth and B does not"
• "B speaks truth and A does not"

Each of these is a conditional event where one person's act of telling the truth (or not) affects the overall situation.

• The probability that A speaks the truth and at the same time, B does not is \( P(T_A \, \text{and} \, F_B) = \frac{4}{5} \times \frac{1}{4} = \frac{4}{20} = \frac{1}{5}\).


• Similarly, the probability that B speaks the truth and A does not is \( P(F_A \, \text{and} \, T_B) = \frac{1}{5} \times \frac{3}{4} = \frac{3}{20} \).

To find the total probability of contradiction, you sum these probabilities. Understanding conditional probability is about seeing how sequences of events play into the overall chances of an outcome, like contradictions.