Let's use the results of the $2012$ presidential election as our x$0.$ Looking up the popular vote totals, we find that our initial distribution vector should be $(0\mathrm{.}5106,0\mathrm{.}4720,0\mathrm{.}0075,0\mathrm{.}0099)$T$.$ Enter the matrix P and this vector x$0$ in MATLAB:

$>>$ P $=[0\mathrm{.}81000\mathrm{.}08000\mathrm{.}16000\mathrm{.}1000$;

$0\mathrm{.}09000\mathrm{.}84000\mathrm{.}05000\mathrm{.}0800$;

$0\mathrm{.}06000\mathrm{.}04000\mathrm{.}74000\mathrm{.}0400$;

$0\mathrm{.}04000\mathrm{.}04000\mathrm{.}05000\mathrm{.}7800]$

$>>$ x$0=[0\mathrm{.}5106$; $0\mathrm{.}4720$; $0\mathrm{.}0075$; $0\mathrm{.}0099]$

According to our model, what should the party distribution vector be after three, six, and ten elections?

What about $30,60,$ and $100$ elections from now? How different are these three results from each other? Summarize what is happening with xk as k gets big.