Question:
A person \(P\) speaks truth in \(75\%\) cases and another person \(R\) in \(80\%\) cases. Then the probability that they are likely to contradict each other in narrating the same event is
A person \(P\) speaks truth in \(75\%\) cases and another person \(R\) in \(80\%\) cases. Then the probability that they are likely to contradict each other in narrating the same event is
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When two people may tell truth or lie independently, contradiction occurs when exactly one tells the truth and the other lies.
Updated On: Jun 26, 2026
- \(\frac{7}{20}\)
- \(\frac{7}{10}\)
- \(0.2\)
- \(0.3\)
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The Correct Option is A
Solution and Explanation
Step 1: Write the given probabilities.
Probability that \(P\) speaks the truth is \[ P(T_P)=\frac{75}{100}=\frac{3}{4}. \] Therefore, the probability that \(P\) lies is \[ P(L_P)=1-\frac{3}{4} =\frac{1}{4}. \] Similarly, the probability that \(R\) speaks the truth is \[ P(T_R)=\frac{80}{100} =\frac{4}{5}. \] Therefore, the probability that \(R\) lies is \[ P(L_R)=1-\frac{4}{5} =\frac{1}{5}. \]
Step 2: Find the cases when they contradict each other.
They contradict each other when \[ (P \text{ tells truth and } R \text{ lies}) \] or \[ (P \text{ lies and } R \text{ tells truth}). \]
Step 3: Calculate the required probability.
Using independent events, \[ P(\text{contradiction}) = P(T_P)P(L_R) + P(L_P)P(T_R). \] Substituting the values, \[ = \left(\frac{3}{4}\right)\left(\frac{1}{5}\right) + \left(\frac{1}{4}\right)\left(\frac{4}{5}\right). \] \[ = \frac{3}{20} + \frac{4}{20}. \] \[ = \frac{7}{20}. \]
Step 4: Final conclusion.
Therefore, \[ \boxed{\frac{7}{20}} \] and the correct option is \[ \boxed{(1)}. \]
Probability that \(P\) speaks the truth is \[ P(T_P)=\frac{75}{100}=\frac{3}{4}. \] Therefore, the probability that \(P\) lies is \[ P(L_P)=1-\frac{3}{4} =\frac{1}{4}. \] Similarly, the probability that \(R\) speaks the truth is \[ P(T_R)=\frac{80}{100} =\frac{4}{5}. \] Therefore, the probability that \(R\) lies is \[ P(L_R)=1-\frac{4}{5} =\frac{1}{5}. \]
Step 2: Find the cases when they contradict each other.
They contradict each other when \[ (P \text{ tells truth and } R \text{ lies}) \] or \[ (P \text{ lies and } R \text{ tells truth}). \]
Step 3: Calculate the required probability.
Using independent events, \[ P(\text{contradiction}) = P(T_P)P(L_R) + P(L_P)P(T_R). \] Substituting the values, \[ = \left(\frac{3}{4}\right)\left(\frac{1}{5}\right) + \left(\frac{1}{4}\right)\left(\frac{4}{5}\right). \] \[ = \frac{3}{20} + \frac{4}{20}. \] \[ = \frac{7}{20}. \]
Step 4: Final conclusion.
Therefore, \[ \boxed{\frac{7}{20}} \] and the correct option is \[ \boxed{(1)}. \]
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