All the components in the state vector need to sum to 1. You're given that component corresponding to state 1 is 0.2, and that the component for state 3 is 0.
That leaves states 2 and 4, for which you're told that the component for state 2 is four times as large. If [tex]x_i[/tex] is the component for state [tex]i[/tex], then you have
[tex]x_1+x_2+x_3+x_4=1\iff 0.2+4x_4+0+x_4=1\implies5x_4=0.8\implies x_4=0.16[/tex]
which means [tex]x_2=4x_4=0.64[/tex]. So the state vector is [tex]\mathbf x=(0.2,0.64,0,0.16)[/tex].
