A can solve 90% of the problems given in a book and B can solve 70%. What is the probability that at least one of them will solve the problem, selected at random from the book?

Let E and F be the events defined as follows:

E = A solves the problems, F = B solves the problem.

Clearly, E and F are independent events such that 

P(E) = 90/100 = 9/10 and P(F) = 70/100 = 7/10

Now,

Required probability = P (E ∪ F)