Problem 35 - Algebra 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

The probability that \(A\) speaks truth is \(4 / 5\) and the probability that \(B\) speaks truth is \(3 / 4\). The probability that they contradict each other when asked to speak on a fact is (a) \(3 / 10\) (b) \(7 / 20\) (c) \(1 / 4\) (d) \(2 / 5\)

Step-by-Step Solution

Answer

The probability that A and B contradict each other is \( \frac{7}{20} \) (option b).

Step 1: Understand the Problem

We need to find the probability that Speakers A and B contradict each other when speaking about a fact. Speaker A speaks the truth with probability \( \frac{4}{5} \). Speaker B speaks the truth with probability \( \frac{3}{4} \). They contradict each other if one speaks the truth and the other lies.

Step 2: Probability of A Speaking Truth

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

Step 3: Probability of B Speaking Truth

The probability that B speaks the truth is \( \frac{3}{4} \). Therefore, the probability B lies is given by: \( 1 - \frac{3}{4} = \frac{1}{4} \).

Step 4: Conditions of Contradiction

A contradiction occurs when A speaks the truth and B lies, or when B speaks the truth and A lies. We calculate the probability for each.

Step 5: Calculate Probability of A Truth, B Lies

The probability that A speaks the truth and B lies is \( \frac{4}{5} \times \frac{1}{4} = \frac{4}{20} = \frac{1}{5} \).

Step 6: Calculate Probability of B Truth, A Lies

The probability that B speaks the truth and A lies is \( \frac{3}{4} \times \frac{1}{5} = \frac{3}{20} \).

Step 7: Total Probability of Contradiction

Add the probabilities of the two contradicting situations: \( \frac{1}{5} + \frac{3}{20} = \frac{4}{20} + \frac{3}{20} = \frac{7}{20} \).

Step 8: Conclusion

The probability that A and B contradict each other when asked to speak on a fact is \( \frac{7}{20} \). Choose option (b).

Key Concepts

Contradiction in Probabilities

When discussing probabilities, contradictions can occur in scenarios where two people, A and B in this case, give opposing statements about the same fact. To quantify their likelihood of contradicting one another, we need to consider situations where one tells the truth and the other lies. This forms the basis of a contradiction in probabilities.
Imagine two events happening in this context. Event 1 is A telling the truth while B lies. Event 2 is B telling the truth while A lies. Each of these occurrences has its own probability, and the combined probability of contradiction is the sum of these individual probabilities. This reflects the likelihood of both scenarios yielding a contradiction.

• Probability that A is truthful while B lies: This happens if A's factual accuracy leads and B's claim fails. Calculated as a product of A's truth speaking probability and B's lying probability.
• Probability that B is truthful while A lies: This is when B's statement holds true against A's incorrectness, found similarly by multiplying B's truth probability and A's lying chance.

Ultimately, when assessing the probability of contradiction, it's about understanding these counter scenarios and adding up their probabilities.

Truth and Lies Probability

In situations where probability plays a role in honest and dishonest expressions, knowing the probability that someone speaks the truth helps us predict likely outcomes. In our case, we have Speaker A, who tells the truth with a probability of \( \frac{4}{5} \), and Speaker B, who does so with a probability of \( \frac{3}{4} \). Understanding these probabilities sheds light on how frequently each individual might communicate accurately.

First, let's consider why knowing these probabilities is essential:

• If A has a higher truth-telling probability, then A is more reliable than B; similar logic applies if the roles are reversed.
• When considering combinations of truth and lies, understanding each individual's truth propensity allows for calculating scenarios like both speaking truth, both lying, or cross scenarios where one lies and the other doesn't.

The essence here is in breaking down each person's likely choices between truth and lying. Only then can we truly predict which outcomes are most probable, not only for individual truths but also for contradictions or agreements between them.

Probability in Decision Making

Understanding probability is vital when making decisions based on potential truth or lies in statements. In scenarios where decisions rely on information from different sources, as in our example with Speakers A and B, it's crucial to recognize how these probabilities can influence outcomes.

Here's how probability can steer decision-making:

• An action dependent on truth statements should consider the probabilities of each source speaking truthfully versus lying. This means recognizing which source provides more reliable data and accordingly weighting decisions.
• By evaluating the chance of contradictions, one can gauge the risk associated with taking a stand on information provided by A and B, especially when they have opposing statements.
• Probability helps in assessing the most likely outcomes, allowing decisions to be based on quantified risks rather than assumptions. Hence, improving accuracy in forecasting or planning strategies.

In essence, probability provides a structured way to navigate decision-making, ensuring conclusions are derived from logically evaluated possible outcomes rather than mere intuition or guesswork.