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:

<img src="https://i.sstatic.net/U0Uov.jpg" width="400" title="complex">

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.