Cantelli inequality ($P(X-\mathbb{E}X\ge a)\le {\sigma }^2/({\sigma }^2+a^2)$)

For a finite-variance random variable, shift its centred version by $b>0$ and apply Markov inequality to the square. Minimizing $({\sigma }^2+b^2)/(a+b{)}^2$ at $b={\sigma }^2/a$ proves the bound. A two-point distribution attains equality. If the variance is zero, the centred variable vanishes almost surely.