There are 50 telephone lines in an exchange. The probability that any one of them will be busy is 0.1, then the probability that all the lines are…

A can solve 90% of problems: $P(A \text{ solves}) = 0.90$P(A solves)=0.90

B can solve 70% of problems: $P(B \text{ solves}) = 0.70$P(B solves)=0.70

A problem is randomly selected from the book.

The problem is solved if at least one of them (A or B or both) can solve it.

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For the probability that A or B solves the problem:

$$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$$P(A or B)=P(A)+P(B)−P(A and B)

The overlap $P(A \text{ and } B)$P(A and B) is subtracted to avoid double counting cases where both solve it.

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Since A and B work independently:

$$P(\text{Both solve}) = P(A) \times P(B)$$P(Both solve)=P(A)×P(B)

$$P(\text{Both solve}) = 0.90 \times 0.70$$P(Both solve)=0.90×0.70

$$P(\text{Both solve}) = 0.63$$P(Both solve)=0.63

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The probability that the problem is solved:

$$P(\text{Problem is solved}) = P(A) + P(B) - P(\text{Both})$$P(Problem is solved)=P(A)+P(B)−P(Both)

$$P(\text{Problem is solved}) = 0.90 + 0.70 - 0.63$$P(Problem is solved)=0.90+0.70−0.63

$$P(\text{Problem is solved}) = 1.60 - 0.63$$P(Problem is solved)=1.60−0.63

$$P(\text{Problem is solved}) = 0.97$$P(Problem is solved)=0.97

Therefore, the probability that the problem is solved is $0.97$0.97.