Hello,

Let $x\in[0;1]$ and $(B_i)_i$ be events defined by $P(B_i)\leq x, \forall i$. Furthermore, this inequality is independent of the other events $B_i$ but the events are not necessarily independent.

I want to upperbound the probability of $A_k = (B_1\cup B_2)\cap (B_2\cup B_3)\cap\cdots\cap (B_k\cup B_{k+1})$. The first terms give (I simplified each $A_i$):

$$P(A_1)\leq 2x$$

$$P(A_2)=P((B_1\cap B_3) \cup B_2)\leq x^2 + x$$

$$P(A_3)=P((B_2\cap B_3)\cup(B_2\cap B_4)\cup(B_1\cap B_3))\leq 3x^2$$

What's the upperbound for any $k$ ?

Thank you very much