A certain professor gives exams that contain an infinite
number of problems. In order to solve each problem, the student
must have solved all of the preceding problems perfectly, since
the professor stops grading an exam as soon as he finds a mistake.
Each student has probability p of getting any particular problem
right, independently of the other students, and independently of the
problem number. The professor gives an A+ to the student who solves
the most problems, provided that only one student has the maximum score.
Otherwise, nobody in the class gets A+.
Write down an expression for the probability that an A+ is given
when n students take the exam. Justify briefly. (Your expression
can be left in the form of a summation, as there appears no "closed form"
for the probability in question.)