Cumulative probability formula - majority of trials

#1
I posted a question below (28 June) on forward error correction which got no replies because it was too naive. I've found the answer I think which is to calculate the probability that a majority of n trials are successful when probability of success of each is p. I want to check my cumulative probability formula for this which I have attached as a pdf. The quantity m is the 'majority' of n which I suppose I could write as Floor(n/2 + 1) but that makes it rather cumbersome. Can someone check that this expression is correct and maybe suggest a symbol relating m to n. I can't find a mathematical symbol for 'majority'.