Is probability of finding electron changing here?
I swear I read the question many times, but I still not quite sure what it is about.  So I will try to clarify some things hoping that they are relevant.



First off, any complex multiple of a wave function indeed describes exactly the same state of a system. This directly follows from the postulate of quantum mechanics which states that the expectation value of an observable $A$ represented by a self-adjoint operator $\hat{A}$ is given as follows,
$$
\langle A \rangle
=
\frac{\langle \Psi | \hat{A} | \Psi \rangle}{\langle \Psi | \Psi \rangle} \, ,
$$
where $\Psi$ is the wave function. It is indeed easy to show that given $\Psi$ we can multiply it by any complex number $c$ and this would not change the expectation value of any observable,
$$
\langle A \rangle
=
\frac{\langle c \Psi | \hat{A} | c \Psi \rangle}{\langle c \Psi | c \Psi \rangle} 
=
\frac{c^{*} c \langle \Psi | \hat{A} | \Psi \rangle}{c^{*} c \langle \Psi | \Psi \rangle}
=
\frac{\langle \Psi | \hat{A} | \Psi \rangle}{\langle \Psi | \Psi \rangle} \, ,
$$
since $c^{*} c$ in numerator and denominator perfectly cancels each other. Also note that the following definition of the expectation value
$$
\langle A \rangle = \langle \Psi | \hat{A} | \Psi \rangle \, ,
$$
holds true only for a normalized wave function $\Psi$ and trivially follows from the above mentioned more general definition.



Secondly, for the simplest case of one particle in one spatial dimension, if one would like to interpret the wave function $\Psi(x,t)$ as a probability amplitude that the particle is at $x$, and consequently, it square modulus, $ \left|\Psi(x, t)\right|^2 = {\Psi(x, t)}^{*}\Psi(x, t)$ as the probability density that the particle is at $x$, he/she have to normalize it as follows,
$$
\int\limits_{-\infty}^\infty d x \, |\Psi(x,t)|^2 = 1 \, .
$$
This is required because one of the axioms of the probability theory requires that


  the probability that some elementary event in the entire sample space
  will occur is 1.


The sample space is just a set of all possible outcomes in an experiment, which in our case is the set of events of finding a particle at any possible position, i.e anywhere between $-\infty$ and $+\infty$. Thus, in our case the probability of finding the particle anywhere between $-\infty$ and $+\infty$ have to be 1, and consequently, the wave function has to be normalized. Otherwise its interpretation as a probability amplitude would be inconsistent with the probability theory.



Hope it helps.