We are given a single error correcting Hamming Code $H(n,k)$ for a Binary Symmetric Channel $BS{C}_p$ having crossover probability $p\frac{1}{2}.$ Answer the following:

$6+3+(3+3)=15$

$($a$)$ Compute the probability $P(n,t)$ of $t,0tn$ errors in an $n-$bits code.

$($b$)$ Compute the probability $P_{\text{c}\text{o}\text{r}\text{r}\text{e}\text{c}\text{t}}(n,p)$ of correctly receiving a code.

$($c$)$ Compute $P_{\text{c}\text{o}\text{r}\text{r}\text{e}\text{c}\text{t}}(n,p)$ for

i$.n=7$ and $p=0\mathrm{.}01,p=0\mathrm{.}1,$ and $p=0\mathrm{.}2$

ii$.n=11$ and $p=0\mathrm{.}01,p=0\mathrm{.}1,$ and $p=0\mathrm{.}2$