Consider the discrete random variable $Y$, the continuous random variable $X$, and a constant $c$. The goal is to find $$P(|X - E_Y[h(y)]| < c),$$ when we are only given $P(y)$, function $h(y)$, and $P(|X-h(y)| < c|Y=y)$ for all possible values of y in domain of Y.
UPDATE: I have tried to write $P(|X - E_Y[h(y)]| < c)$ based on density function of $f(x,y)$ but this didn't get me so much far as the boundaries of integral doesn't let me to take the summation inside : $$P(|X - E_Y[h(y)]| < c) = \sum_y\int_{-c+E_Y[h(y)]}^{c+E_Y[h(y)]}f(x,y)dx$$