An examination is conducted entirely by multiple choice questions. There are 100 questions, and 3 answers are shown for each, of which 1 is correct. A student passes if he or she chooses 45 or more correct answers. A student atempts this examination without knowing anything about the subject, and guesses all the answers( you may assume the answers are a random choice.)
a) what is the probability that the student passes at the first atempt? (Leave the answer in the form of a sum)
b) If the student fails, he or she retakes the examination again and again until he or she passes. The student always chooses the answers randomly. What is the probability distribution of the number of attempts at the examination made by the student? ( Leave the answer as an expression)
c) The authorities conducting the examination realized that it was possible to pass by guessing answers, and they decided to fix a maximum number of attempts a student may make. What number of attempts should they allow if they require the total probability of passing by guessing to be less than 2 %. ( again, leave your answer in the form of " N < some expression. ")