Accept-or-fallback mixture identity ($\mathbb{P}(X=x)=p_x{a}_x+{\sum }_y{p}_y(1-a_y)K(y,x)$)

Let a discrete random variable $Y$ have probability mass function $p_y$. Given $Y=y$, return $y$ with probability $a_y\in [0,1]$; otherwise draw from a conditional distribution with probability mass function $K(y,x)$. The law of total probability gives the displayed identity. Its total mass is ${\sum }_y{p}_y[a_y+(1-a_y){\sum }_{x}K(y,x)]=1$. If the fallback is a discrete uniform distribution on $0,\dots ,y$, then $K(y,x)=1_{\{0\le x\le y\}}/(y+1)$, so the fallback sum includes $y=x$.