= Solution
Put $r=p/(E+m)$ and assume $p>0$. The common <plane wave> has unit <Complex modulus>[modulus]. The component <norm> is
$$
\psi^\dagger\psi=|N|^2[(1+r)^2+4(1-r)^2+(1-r)^2+4(1+r)^2]=10|N|^2(1+r^2).
$$
For unit <spinor> <probability density> choose
$$
\boxed{N=\frac1{\sqrt{10(1+r^2)}}=\sqrt{\frac{E+m}{20E}},}
$$
where $E^2=p^2+m^2$ was used. This is <spinor> <normalization>, not square <normalization> of a <plane wave> over infinite space; a box or <Wave packet>[wavepacket] gives the corresponding spatial <normalization>. If the convention is instead $\psi^\dagger\psi=2E$, use $N=\sqrt{(E+m)/10}$.
For <momentum> along $+z$, the <Helicity>[positive-helicity] components are the first and third, while the second and fourth have negative <helicity>. Hence
$$
\boxed{P(h=+1/2)=2|N|^2(1+r^2)=\frac15,\qquad P(h=-1/2)=8|N|^2(1+r^2)=\frac45.}
$$
Both are independent of <speed>. In the source's chiral block order the exact <Chirality>[left-chiral] weight is $1/2+3p/(10E)$, tending to $4/5$ at high <energy>; <helicity> and <chirality> <probabilities> agree only in that limit.
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