A speaks the truth 40% of the time, and B speaks the truth 50% of the time. The probability that they contradict each other while narrating an incident is:
Given: \[ P(A) = 40\% = \frac{4}{10}, \quad P(B) = 50\% = \frac{5}{10}. \] We are asked to compute: \[ P(A) \cdot P(\bar{B}) + P(B) \cdot P(\bar{A}). \] First, calculate \( P(\bar{A}) \) and \( P(\bar{B}) \): \[ P(\bar{A}) = 1 - P(A) = 1 - \frac{4}{10} = \frac{6}{10}, \quad P(\bar{B}) = 1 - P(B) = 1 - \frac{5}{10} = \frac{5}{10}. \] Now substitute the values into the given expression: \[ P(A) \cdot P(\bar{B}) + P(B) \cdot P(\bar{A}) = \frac{4}{10} \times \frac{5}{10} + \frac{5}{10} \times \frac{6}{10}. \] Simplifying: \[ = \frac{20}{100} + \frac{30}{100} = \frac{50}{100} = \frac{1}{2}. \] Thus, the value of the expression is \( \frac{1}{2} \).
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