The probability of electron occurring in fermi level is half. In case of intrinsic semiconductors the fermi level lies halfway between the valence band and conduction band i.e. in the band gap. How can electrons occur in the band gap? Correct me if am wrong anywhere. Thanks for your replies!!
The Fermi-Dirac distribution \$F(\epsilon)\$ gives the probability that (at thermodynamic equilibrium) an electron will occupy a state having energy ϵ.
That is, the Fermi-Dirac distribution doesn't give the probability of the existence of an electron at a certain energy. Rather, if there is a state at that energy, it gives the probability of the state being occupied.
So if there are no states, there won't be any electrons regardless of the Fermi level.
The 50% probability in the case of semiconductors really should be viewed as "what is the average energy of an electron in this system". Given you have certain statistics and populations in both the valence and conduction band it makes sense that the average is indeed 50%. An average of two equally populated distributions is also 50% (which is mid band-gap).