$10.$ Consider a frame transmitted across a noisy channel. Each time the frame is transmitted, the probability that it will arrive without error is $0\mathrm{.}75.$ What is the mean number of transmissions needed to deliver the frame, without error, across the channel? Show your work.

The loss rate L is $0\mathrm{.}25.$ The expected number of transmissions is $1/(1$ L$)=1/(10\mathrm{.}25)=1/0\mathrm{.}75=4/31\mathrm{.}33$

$11.$ Error detection with CRC

$($a$)$ Consider a CRC code with a generator polynomial of G$($x$)=$ x$4+$x$3+1.$ Compute the checksum generated on the following short data frame: $10001011.$ You must show your work.