Upper bound for probability

Question

Given two i.i.d. random variables $X,Y$ that satisfy the following condition:

1. $P(|X|<0.5)<a$ and
2. $P(|Y|<0.5)<b$

How can I derive an upper bound for the following probability $$P\left(\middle|\frac{X+Y}{2}\middle|<0.5\right)$$ as a function of $a$ and $b$? Thanks.