IMC 2011 · Day 1 · P4

Let $\Omega = \bigcup\limits_{i=1}^{n} A_i$. Consider a random subset $X$ of $\Omega$ which chosen in the following way: for each $x \in \Omega$, choose the element $x$ for the set $X$ with probability $t$, independently from the other elements.

Then for any set $C \subset \Omega$, we have $$P(C \subset X) = t^{|C|}.$$ By the inclusion-exclusion principle, $$\begin{align*} P \bigl( (A_1 \subset X) \text{ or } (A_2 \subset X) \text{ or } \dots \text{ or } (A_n \subset X) \bigr) &= \\ = \sum_{k=1}^{n} \sum_{1 \le i_1 < i_2 < \dots < i_k \le n} (-1)^{k-1} P \bigl( A_{i_1} \cup A_{i_2} \cup \dots \cup A_{i_k} \subset X \bigr) &= \\ = \sum_{k=1}^{n} \sum_{1 \le i_1 < i_2 < \dots < i_k \le n} (-1)^{k-1} t^{|A_{i_1} \cup A_{i_2} \cup \dots \cup A_{i_k}|}. & \end{align*}$$ The probability $P \bigl( (A_1 \subset X) \text{ or } \dots \text{ or } (A_n \subset X) \bigr)$ is a nondecreasing function of the probability $t$.