Probability of Poisson process interval containing nearby points? I came across a problem recently which seems like should be a straightforward application of the Possion distribution but it isn't quite coming together for me. Given a Poisson process with rate $\lambda$ what is the probability that, within the unit interval [0,1], at least two events occur within $1/4$ units of each other? (i.e. events occuring at t=0.8 and 0.9 would be a success, but events occurring at t=0, t=0.3, and t=0.9 would be a failure)
I thought about using each of the sub-intervals of size $1/4$ each with a probability $1-(P(k=0)+P(k=1))$ of containing two events; however this approach seems doomed to failure as the sub intervals are not independent.
Thanks for the help!
 A: The distance $T_1$ from $0$ to the first arrival in the Poisson process had an exponential distribution with expectation $1/\lambda.$ The distance $T_2$ from there to the second arrival has that same distribution, and $T_2$ is independent of $T_1.$ Similarly $T_3,T_4,\ldots$ The event whose probability you seek can then be expressed as$$\begin{align} & \Big[\left( T_2-T_1<\frac 1 4 \text{ and } T_2<1 \right) \text{ or } \left( T_3-T_2<\frac 1 4 \text{ and } T_3 < 1 \right) \\[10pt] \text{ or } & \left( T_4 - T_3 < \frac 1 4 \text{ and } T_4<1 \right) \text{ or } \cdots \cdots\cdots \Big] \end{align}$$
The complementary event is
$$
\begin{align}
& \left( T_2-T_1 > \frac 1 4 \text{ or } T_2>1  \right) \text{ and } \left( T_3-T_2>\frac 1 4 \text{ or } T_2>1 \right) \\[10pt]
\text{and } & \left( T_$ - T_3 > \frac 1 4 \text{ or }  T_3 > 1 \right) \text{ and } \left( T_4 - T_3 > \frac 1 4 \text{ or } T_4 > 1 \right). \text{ [full stop]} 
\end{align}
$$
Here we need only finitely many propositions connected by "and". 
Some work remains after that! But this shows you how to reduce it to an essentially finite problem.
