Suppose I have $n$ trials where each trial consists of $m$ fair coin flips.
What bound can give the following statement to upper bound the number of trials with many heads? For any $\delta > 0$, there exists $n_0$ s.t. if $n > n_0$ w.p. $1-\delta$ there are at most $N$ trials with more than $k$ heads.
This may be a known concentration bound, but I could not find anything quite like this when looking for variations of Hoeffding or Chernoff. Apologies for the vague question.