Question : A speaks the truth 5 out of 7 times and B speaks the truth 8 out of 9 times. What is the probability that they contradict each other in stating the same fact?
Option 1: $\frac{1}{7}$
Option 2: $\frac{1}{9}$
Option 3: $\frac{1}{4}$
Option 4: $\frac{1}{3}$
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Correct Answer: $\frac{1}{3}$
Solution :
Probability of A speaking truth = $\frac{5}{7}$ = P(A)
⇒ Probability of A lying = $1-\frac{5}{7}$ = $\frac{2}{7}$ = P(A')
Probability of B speaking truth = $\frac{8}{9}$ = P(B)
⇒ Probability of B lying = $1-\frac{8}{9}$ = $\frac{1}{9}$ = P(B')
$\therefore$ Required probability
= P(A) × P(B′) + P(A′) × P(B)
= $\frac{5}{7}×\frac{1}{9}+\frac{2}{7}×\frac{8}{9}$
= $\frac{5}{63}+\frac{16}{63}$
= $\frac{21}{63}$
= $\frac{1}{3}$
Hence, the correct answer is $\frac{1}{3}$.
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