Expected Xor

Monisha likes to paint. She has painted $N$ paintings (numbered $1$ through $N$) and wants to choose some subset of these paintings for an exhibition. For each valid $i$, the $i$-th painting has *beauty* $b_i$ and the probability that it will be displayed at the exhibition is $p_i$. Each painting is chosen or excluded from the exhibition independently randomly. The beauty of the resulting exhib