The non-normalized wavefunction of a general qubit is given by: $$|\psi\rangle=A|0\rangle+B|1\rangle.$$ The complex amplitudes $A$ and $B$ can be represented by two arrows in the complex plane:
Now the wavefunction can be multiplied by any complex number $R$ without changing the physics. This will cause the arrows $A$ and $B$ to rotate and shrink/expand together with a fixed angle between them.
Therefore two sets of points will be traced out represented by a circle with area $|A|^2$ and a circle with area $|B|^2$. These represent the sets of possible values for the amplitudes $A$ and $B$.
Thus if we become entangled with the qubit then the probabilities of finding ourselves in set $A$ (measuring $0$) or set $B$ (measuring $1$) are given by: $$P(0)=\frac{|A|^2}{|A|^2+|B|^2}$$ $$P(1)=\frac{|B|^2}{|A|^2+|B|^2}.$$
Does this picture help to understand the origin of probabilities in quantum mechanics?
Correction
Let $$A=R_Ae^{i\theta_A}$$ $$B=R_Be^{i\theta_B}$$ A general normalized wavefunction is given by: $$|\psi\rangle=\frac{1}{(R_A^2+R_B^2)^{1/2}}\large[R_Ae^{i\theta_A}+R_Be^{i\theta_B}\large]$$ Assume that I multiply the amplitudes $A$ and $B$ by $$C=Re^{i\theta}$$ Then the normalized wavefunction becomes $$|\psi\rangle=\frac{1}{R(R_A^2+R_B^2)^{1/2}}\large[RR_Ae^{i(\theta_A+\theta)}+RR_Be^{i(\theta_B+\theta)}\large]$$ $$|\psi\rangle=\frac{e^{i\theta}}{(R_A^2+R_B^2)^{1/2}}\large[R_Ae^{i\theta_A}+R_Be^{i\theta_B}\large]$$ It seems that the only degree of freedom is a phase angle $\theta$ rather than an area as I asserted above.
