Suppose that Alice sends Bob the packets m_{s_{1}}, m_{s_{2}}, \ldots Let d_{i}=\operatorname{dim}\left(\left\{s_{1}, \ldots, s_{i}\right\}\right), with d_{0}=0.
Alice begins sending packets to Bob.
(i) (5 points) Find the exact probability that Bob can recover all n original messages after n packets, and show that this probability is at least \frac{1}{4} for all n.
(ii) (5 points) Show the probability that Bob can recover the n message after n+2 packets is at least \frac{3}{5} for all n \geq 5.