quantum mechanics - Don't electrons have discrete energy levels in semiconductors?

I am having a hard time understanding the relation between the fermi distribution of electrons in a semiconductor, and the fact the electron energy states are discrete.

The fermi distribution is supposed to give us the probability of an energy level being occupied by electrons. It looks like this:

yet electrons are supposed to have discrete energy states, and no electron is supposed to be in the band gap:

I must be missing something here, how is it that electrons have a fairly high probability of being in the band gap ?

Answer

There are two important contributions: First the density of states, which tells you the states, which can potentially be occupied. Then there is the Fermi-Dirac distribution, which tells you, which energies are occupied. The Fermi-Dirac distribution does not include allowed and forbidden states. You must fold it with the density of states, which is then called combined density of states. This function then tells you the distribution of occupied or unoccupied states.