0.1 Question 1 Let q E (0, 1). A machine produces N devices during its life-time, where P (N = n) = cqn, n> 1 P (N = 0) =1-can. n=1 Each device the machine produces is equally likely to be faulty or non-faulty, independently of the other devices. (a) [5 marks] Show that the probability that the machine produces j faulty devices is 2cqj (2- 9) 3+1 for j > 1. CONTENTS You may use without proof that for r E (0, 1), k( k - 1) ... (k -m+ 1)xk-m= m! KO (1 - x)m+1 (b) [4 marks] Given that the machine has produced at least 1 faulty device over its lifetime, what is the probability that the machine has produced at least 2 faulty devices over its lifetime