When trying to estimate the proportion of a sample (eg. proportion of patients undergoing an operation who experience a complication) one starts with the data (number of "successes", number of trials) and a prior distribution of that proportion parameter, typically beta(1,1).
What happens when one knows as sure "as God made little apples" that the proportion who will experience complications is not 50%. What happens if we are confident that the parameter possibilities do not span the spectrum from 0 to 100%. We expect the complication rate to be about 10% or there abouts. 50% would be very very very unlikely. If you want to see more details about my case you can look here but my question is a general one.