Consistency limit equals a log

Let $X_{1},X_{2},…,X_{n}$ be a random sample from $U(1,2)$ distribution and let $Y_{1},Y_{2},…,Y_{n}$ be a random sample from $U(0,1)$ distribution. Suppose that the two random samples are independent. Define $$Z_{i}=\begin{cases}1,&\text{if }X_{i}Y_{i}<1\\0,&\text{otherwise},\end{cases}\quad i=1,2,…,n$$ If $\lim_{n\to\infty}P\left(\left|\frac{1}{n}\sum_{i=1}^{n}Z_{i}-\theta\right|<\epsilon\right)=1$, for all $\epsilon>0$, then $\theta$ equals