A speaks truth in 60% cases and B speaks truth in 70% cases. The probability that they will say the same thing while describing a single event, is

### Concept & Logic Two people will say the same thing (agree) in two scenarios: either they both speak the truth, or they both lie about the event. $$ P(\text{Agreement}) = P(A_{\text{truth}})P(B_{\text{truth}}) + P(A_{\text{lie}})P(B_{\text{lie}}) $$ ### Step-by-Step Solution * Probability A speaks truth: $P(A) = 60\% = 0.6$ * Probability A lies: $P(\bar{A}) = 1 - 0.6 = 0.4$ * Probability B speaks truth: $P(B) = 70\% = 0.7$ * Probability B lies: $P(\bar{B}) = 1 - 0.7 = 0.3$ * They say the same thing if (A speaks truth AND B speaks truth) OR (A lies AND B lies). * Required Probability $= [P(A) \times P(B)] + [P(\bar{A}) \times P(\bar{B})]$ * $= (0.6 \times 0.7) + (0.4 \times 0.3)$ * $= 0.42 + 0.12 = 0.54$ ### Exam Strategy & Shortcut Since the options are already in decimal format, sticking to decimal multiplication from the start ($0.6 \times 0.7 + 0.4 \times 0.3$) is the fastest route. Avoid converting to fractions just to convert back. ### Common Pitfall A common mistake is only calculating the probability that they both speak the truth ($0.6 \times 0.7 = 0.42$) and forgetting that they will also say the same thing if they both decide to lie about the event. ### Final Answer Therefore, the correct answer is **0.54**.