$[10$ marks$]$ Consider a communications path through a packet switching network with bit errors that occur independently with a probability of $P_e={10}^{-6}.$ Packets travelling through the network are a fixed length of $864$ bits consisting of the following fields:

Destination Address $32$ bits

Information Field $800$ bits

$CRC32$ bits

a$)[2$ marks$]$ What is the probability that a packet is lost before arriving at the destination? $($Assume that a packet is lost if the destination address is corrupted during transit. You may assume that the probability that a packet is lost due to queue overflow at switches is negligible.$)$

b$)[2$ marks$]$ What is the probability that a packet arriving at the destination is detected as having errors, assuming that the CRC can detect all errors?

c$)[3$ marks$]$ Assume that $1000$ packets are to be sent. What is the probability that three or more packets need to be retransmitted?

d$)[3$ marks$]$ Assuming that the CRC can detect one$-,$ two$-$ or three$-$bit error patterns, what is the probability that a received packet has errors that are detected?