The television game “What? Where? When?” consists of a team of “experts” trying to solve 13 questions that are thought up and sent in by the viewers of the program. Envelopes with the questions are selected in turn in a random order with the help of a spinning top with an arrow. If the experts answer correctly, they earn a point, and if they answer incorrectly, the viewers get one point. The game ends as soon as one of the teams scored 6 points. Suppose that the abilities of the teams of experts and viewers are equal.
a) Find the mathematical expectation of the number of points scored by the team of experts in 100 games.
b) Find the probability that, in the next game, envelope number 5 will come up.