I did something similar once when looking at morning line as predictor for winner as opposed to final odds. Final odds, as should be expected did about 3% better than morning line, which is statistically significant considering the size of the sample...thus a partial answer to your question would be very frequently, but the public still knows better than the oddsmaker in the long run.
In 2004, all tracks, the ML favorite was the post-time favorite approx. 65% of the time, which includes ties of either the ML or the post-time odds. (i.e. if there were co-favorites in the odds, and either one of them was the ML favorite then that counts, and vice-versa)
. (Sorry couldn't resist - now back to being on topic)