An exam is made up of three trigonometry problems, two algebra problems and five geometry problems. Martin is able to solves trigonometry problems with probability \(p_1 = 0.2\), geometry problems with probability \(p_2 = 0.4\), and algebra problems with probability \(p_3 = 0.5\). To get a \(B\), Martin needs to solve at least five problems, where the grades are as follows \((A+, A, B, C, D)\).
a) With what probability does Martin solve at least five problems?
Martin decided to work hard on the problems of any one section. Over a week, he can increase the probability of solving the problems of this section by 0.2.
b) What section should Martin complete, so that the probability of solving at least five problems becomes the greatest?
c) Which section should Martin deal with, so that the mathematical expectation of the number of solved problems becomes the greatest?