Hello,

Let say you have events $A_1, A_2, ..., A_r$. Each event has a probability $p$ to occur and the events are independent. Let $b$ be an integer with $b\leq r$.

Compute the probability of the event :

$$ \cap_{i=1}^{r-b+1} \cup_{j=0}^{b-1} A_{i+j} = (A_1\cup A_2\cup ...\cup A_b) \cap (A_2\cup A_3\cup ... \cup A_{b+1}) \cap ... \cap (A_{r-b+1}\cup ... \cup A_r)$$

If you develop this expression, we find many reductions and simplifications but it seems hard, at least to me, to find the exact probability for big values of $r$ even though events are independent and with the same probability.