What do scientists use to predict the locations of electrons in atoms?
Algebra and calculus: Heisenberg's uncertainity principle
According to Werner Heisenberg, it is not possible to find out the location of an electron moving with high speed with precision.
Heisenberg's uncertainity principle is why they have to predict, I suggest reading further into that, as it's quite detailed.
Algebra, calculus and a cheeki bit of good ol' probability is used to guess where it's flying about
Their main tool is the Schrödinger equation.
The Schrödinger equation is a complicated mathematical equation that describes the allowed energy levels of an electron and the probabilities of finding an electron in one of the regions corresponding to those energy levels.
This gives us our familiar pictures of
The orbital boundaries are not sharp like in the picture.
Rather, they are the 90 % probability boundaries — the boundaries within which you can expect to find the electron 90 % of the time.
Then they use rules like the Pauli exclusion principle and Hund's rule to decide which orbital an electron will be in.
Finally, they use other physics formulas to decide just how big the orbital is.