The electron density used in density functional theory for a system of $N$ electrons with wavefunction $\psi$ is defined as
$$\rho(r)=N\int \Psi^*(r,r_2,\dots r_N)\Psi(r,r_2,\dots r_N) d^3r_2\dots d^3r_N$$
The interpretation of this is given as the probability of finding one of the $N$ electrons in the volume element $d^3r$. The following property also holds: $$\int \rho(r)d^3r=N$$ I do not understand this, if $\rho(r)$ is the probability density with aforementioned interpretation, its integral over all space should simply mean: "The probability of finding an electron in all space" and that should be just $1$, not $N$. How can the probability of finding an electron at any point in all space be greater than 1?
Source: A Chemist’s Guide to Density Functional Theory. Second Edition Wolfram Koch, Max C. Holthausen