$(30$ points$)$ Suppose that three independent users, Alice, Bob, and Charlie, are seeking

access to a shared medium using slotted Aloha with binary exponential backoff sub$-$

ject to maximum and minimum limits of the probability pmax $=0\mathrm{.}8$ and pmin $=0\mathrm{.}1.$

Suppose that all three nodes are backlogged, and at slot n $1$ the probabilities the

three nodes transmit packets individually are pa $=0\mathrm{.}6,$ pb $=0\mathrm{.}3,$ and pc $=0\mathrm{.}5,$

respectively.

$($a$)$ What are the possible values of pa at slot n$?$ What are the probabilities associ$-$

ated with each possible value?

$($b$)$ What are the possible values of pc at slot n$?$ What are the probabilities associ$-$

ated with each possible value?