[kgwgk]

On a log-odds (logistic) scale, though, we get something that resembles a normal (a logistic distribution).

[eru]

Yes, a uniform distribution in the space of probabilities between (0,1) would give you something bell-curve shaped in the log-odds scale.

The log-odds scale also makes it clear why probabilities 0 and 1 are very special: they only show up as limits in the log-odds scale.