I have a set of independent events $\{B_1,B_2,...,B_k\}$ where $k\leq n$ for some integer $n>0$.
I know the following probabilities
$$ p(B_i) = (1-r^{\frac{n}{n-i}})\\ p(\overline{B_i} \cap B_j ) =p(\overline{B_i}) p(B_j) = r^{\frac{n}{n-i}}(1-r^{\frac{n}{n-j}})\\ p(B_i \cap B_j ) =p(B_i) p(B_j) = (1-r^{\frac{n}{n-i}})(1-r^{\frac{n}{n-j}}) $$
for some $0<r<1$. Is there an easy way to compute the following probability?
$$ p(B_1 \cup B_2\cup B_3\cup...\cup B_{k-1}\cup \overline{B_k}) $$
I would use the Inclusion-Exclusion Principle to compute it $$ p(B_1 \cup B_2\cup B_3\cup...\cup B_{k-1}\cup \overline{B_k}) = \sum_{r=1}^{k} (-1)^{r+1}\left[\sum_{i_1<i_2<...<i_r<k}\prod_{j=i_1}^{i_r}p(B_j)+\sum_{i_1<i_2<...<i_{r-1}<k}p(\overline{B_k})\prod_{j=i_1}^{i_{r-1}}p(B_j)\right]$$
but I suspect it will just be a mess to derive a general formula. Any advice?