Everyone Lies Sometimes

If $P(A)$ = Prob. that A speaks truth And P( $A^c$ )= Probability that $A$ lies. Similarly for $P(B)$ and P( $B^c$ ).

Hence when they contradict each other it means both of them are true or false. Hence P(Contradiction)= $P ( A^c \cap B) + P ( A \cap B^c)$ .
Now as these events are independent, $P(C)= P (A^c) \times P(B) + P(A) \times P(B^c) = 0.66$

Let's first figure out when their answers won't contradict each other. That will happen when they both speak the truth or when they both don't speak the truth. The probability that they will both speak the truth is $0.3\times 0.9=0.27$ . On the other hand, the probability that they both won't speak the truth is $(1-0.3)\times (1-0.9)=0.07$ . Hence, the probability that their answers won't contradict each other is $0.27+0.07= 0.34$ . Finally, the probability that their answers will contradict each other is $1 - 0.34 =$ $\boxed{0.66}$ .

P(contradict each other)
= P(truthful A & lying B) + P(truthful B & lying A)
= 0.3 x (1 - 0.9) + 0.9 x (1 - 0.3)
= 0.3 x 0.1 + 0.9 x 0.7
= 0.03 + 0.63
= 0.66
= 66%