In the Serre spectral sequence, is it true that we can replace homology by reduced homology? That is: If $f:X\rightarrow B$ is a Serre fibration,with $F$ the fiber, then if $\tilde E^2_{pq}=\tilde H_p(B,\tilde H_q F) \longrightarrow \tilde H_{p+q}(X)$?

I think it is fine: we use complexes involving "augmentation", then filtration, do the usual things in the spectral sequence, and finally we get a sequence that converges to the homology of the original complex. Only now it becomes reduced homology. But I am not precisely sure.

I ask this question because I want to know the answer of another problem (asked by me) "homology dimension of mapping class group of surface with boundary". (I am sorry I don't know how to insert a link). I need some help for that problem. Thanks!