See attached file.
I'll denote spin in the +z direction by (1,0) and in the -z direction by (0,1).
In Quantum Mechanics, if a particle is in some state |p> the probability that after some measurement you find that this particle in some state |q> is given by the absolute value squared of the inner product:
So, if the normalized spin wavefunction is (psi1, psi2), then that means that when you measure the z-component of the spin, the probability that you find that the spin is in the +z direction is:
|(psi1, psi2) dot (1,0)|^2 = |psi1|^2
A detailed solution is given.