Chef at the Food Fair

A food fair is being organised in Chef's town. $N$ booths were set up in a straight line and numbered $1$ through $N$. For each $i$ ($1 \le i \le N$), the probability with which the food served at booth $i$ will cause food poisoning at the beginning of the day is $P_i$. Over the course of the day, the Chef observes that certain booths might decide to offer less unhealthy food based on feedback fr