If we have a beam of spin-1 particles and let them pass through a Stern-Gerlach apparatus (oriented along z-axis, we get three output beams. Suppose we now take only the $+\hbar$ beam and pass it thorugh a Stern-Gerlach apparatus oriented along x-axis, we again get three states and we expect them to have equal probabilities (similar to spin-1/2 particles). Instead, we get beams with the following probabilities, $$P_{+\hbar} = \frac{1}{4} \\ P_{0\hbar} = \frac{1}{2} \\ P_{-\hbar} = \frac{1}{4}.$$ This is contrary to the case of spin-1/2 particles, where when we conduct a similar experiment, we get $+\hbar$ and $-\hbar$ beams with equal probabilities. Where does this non uniformity in the resulting beams arise from for the case of spin-1 particles?

Refer. Spin-1 System, Chapter 2, Quantum Mechanics by David McIntyre http://depts.washington.edu/jrphys/ph248A11/qmch1.pdf

If we have a beam of spin-1 particles and let them pass through a Stern-Gerlach apparatus (oriented along z-axis, we get three output beams. Suppose we now take only the $+\hbar$ beam and pass it thorugh a Stern-Gerlach apparatus oriented along x-axis, we again get three states and we expect them to have equal probabilities (similar to spin-1/2 particles). Instead, we get beams with the following probabilities, $$P_{+\hbar} = \frac{1}{4} \\ P_{0\hbar} = \frac{1}{2} \\ P_{-\hbar} = \frac{1}{4}.$$ This is contrary to the case of spin-1/2 particles, where when we conduct a similar experiment, we get $+\hbar$ and $-\hbar$ beams with equal probabilities. Where does this non uniformity in the resulting beams arise from for the case of spin-1 particles?

Refer. Spin-1 System, Chapter 2, Quantum Mechanics by David McIntyre http://depts.washington.edu/jrphys/ph248A11/qmch1.pdf

Edit. (As suggested in the comments) I am completely comfortable with the mathematical formalism and deriving the probabilities. What I am interested in is a physical reasoning (if it exists) for why the probability is not equal for the three beams.