I'm confused about why the Fermi level is located inside the band gap in semiconductors because it's defined to be the energy level where an electron has a 50% chance to be, but there can't be electrons in the band gap. I now think they mean it's the energy where the Fermi-Dirac distribution reaches 50% but now I'm wondering what this distribution even means when there is a band gap. It's a distribution that gives the occupation probability for a certain energy (given a certain temperature). But why isn't the occupation probability zero in the band gap? Or does it give the occupation in case there was no band gap?
In any case, I was wondering how to interpret the picture below. On the left, the valence band is completely occupied. In the middle, there are a lot of states with a lower occupation probability in the valence band compared to the left, but very few in the conduction band with a higher (but still very small) occupation probability. But it can't be the case that there are suddenly less states being occupied because there are still as many electrons as before. So what am I getting wrong? And how can I interpret the Fermi-Dirac distribution better?
Thanks in advance.