Problem $6(10$ points$).$ Suppose, there are two modes of transportation available between two cities, Bus and Auto, with the following utility functions: $U_{\text{B}\text{u}\text{s}}=-1\mathrm{.}0-0\mathrm{.}01x_1-0\mathrm{.}2x_2,U_{\text{A}\text{u}\text{t}}=-0\mathrm{.}5-0\mathrm{.}05x_1-0\mathrm{.}1x_2,$ where $x_1$ is the travel time $($ min $)$ and $x_2$ is the out$-$of$-$pocket cost $($dollars$)$ with the following values: Bus: $x_1=15,x_2=2\mathrm{.}5.$ Auto: $x_1=8\mathrm{.}0,x_2=0\mathrm{.}10.$ With these values of $x_1$ and $x_2,$ determine:

$($a$)$ the utility of each mode of transportation, and

$($b$)$ the probability of selecting each mode of transportation.