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I want to prove the estimate

\begin{equation*} P(|A+B|>c)\leq P(|A|>c/2)+P(|B|>c/2), \end{equation*}

where $A$ and $B$ are random variables and $c>0$. Can anyone help?

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2 Answers 2

up vote 2 down vote accepted

Hint: Use the following: If $|A|\leq c/2$ and $|B|\leq c/2$, then $|A+B|\leq c$. It might be useful to negate this statement and then use the subadditivity of the probability measure.

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Yes, it directly follows from the union bound.

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