Four computers are connected in a network according to the following diagram, where each line rep- resents a connection. Each connection has a probability p of being broken at any time, independent of any other connections. A computer is isolated if all connections to it are broken.

(a) What is the probability that computer 1 is isolated? [5 marks] (b) What is the probability that both computers 1 and j are isolated for j = 2, 3, 4? [5 marks]

(c) What is the probability that no computer is isolated in this network? [5 marks]

(When answering the questions, writing down the final answer will not be sufficient to receive full marks. Please show all calculations unless otherwise specified. Also define any events and notation that you use in your solutions.)

Four computers are connected in a network according to the following diagram, where each line rep- resents a connection. Each connection has a probability p of being broken at any time, independent of any other connections. A computer is isolated if all connections to it are broken. A 1 2 (a) What is the probability that computer 1 is isolated? [5 marks] (b) What is the probability that both computers 1 and j are isolated for j = 2, 3,4? [5 marks] (c) What is the probability that no computer is isolated in this network? [5 marks]