# Combinatoric problem with the development of intersection of union of events

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.

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This event means that in the sequence of outcomes $A_1, ..., A_r$ you don't have $b$ adjacent falses. Suppose $b \le t$. Let $r-t$ be the index of the last true event in that sequence. Then $0 \le t < b$ you get this sequence from an $r-t-1$ long sequence with this property by appending t falses and a true. Thus, if you name the probability of this event a(r), you have the recurrence $$a(r) = \sum_{0 \le t < b} p(1-p)^ta(r-t-1),$$ and the starting conditions $a(r) = 1$ if $r < b$.
Fix any errors in the above argument, then try to solve the recurrence. For any fixed $b$, this is a linear recurrence, so it has an explicit solution. With $b$ as a parameter, it might be hard, though you could still ask for an approximation. The Concrete Mathematics book may help.