If stuff like this interests you, I would suggest reading about Concentration Inequalities [1][2]

Concentration inequalities unify both of these inequalities and provide tools to make new ones for whatever situations you have.

1. https://www.stat.berkeley.edu/~mjwain/stat210b/Chap2_TailBounds_Jan22_2015.pdf
2. http://84.89.132.1/~lugosi/anu.pdf