$48.$ Suppose Ethernet physical addresses are chosen at random $($using true random bits$).$

$($a$)$ What is the probability that on a $1024-$host network, two addresses will be the same?

$($b$)$ What is the probability that the above event will occur on one or more of $220$ networks?

$($c$)$ What is the probability that, of the $230$ hosts in all the networks of $($b$),$ some pair has the same address?