Let $x_1, \ldots, x_n$ be **possibly dependent** random variables, each taking values $x_i \in \{0, 1, 2\}$. Suppose further that in every outcome the number of random variables that equal 2 is exactly 1. Now for each $i \in \{1, \ldots, n\}$ define
$$
f_i = \begin{cases}
\Pr[x_i = 2 \mid x_i \geq 1] & \text{if } x_i \geq 1\\
0 & \text{if } x_i =0
\end{cases},
$$
and let
$
f = \sum_i f_i.
$

My question is how large can the variance of $f$ be? My conjecture is that we should be able to bound it by $O(1)$ but don't know how to prove this.

Note: In case it helps, it is easy to prove that $E[f] = 1$: $$ E[f] = \sum_i E[f_i] = \sum_i \Pr[x_i \geq 1] \times \Pr[x_i = 2 \mid x_i \geq 1] = \sum_i \Pr[x_i = 2] = 1, $$ where the last equality comes from our initial assumption that in all outcomes exactly one of the $x_i$'s equals 2.