These are not the same in general (mean and mode respectively). Which did you want?the expectation value of event A (the most probable number of times event happens)
I flip a symmetrical coin n times and I'm interested to know the expectation value of event A (the most probable number of times event happens), which is defined as fliping head m times in a row (m<=n). It is important to know that, for example m=10, fliping 20heads in a row represents two A events, not 11 events, also 19heads in a row represents only 1 event.