Calculating the probability that two or more clocks ring

Question

Given a set of iid random Variables $X_n\sim \exp(\lambda)$.
We have a (countably) infinite number of clocks $c_1,c_2,...$.
Let $t=T$ be the maximum time.
When our time starts (at $t=0$) the first clock begins to run and rings on the interval $(0,t_1)$ with a probability of $1-\exp(-\lambda t_1)$. When the first clock rings, the second starts to run and independently from the first clock this one rings on the interval $(t_1,t_2)$ with a probability of $1-\exp(-\lambda *(t_2-t_1))$. This continues like this with the third clock , fourth clock... until we reach time $t=T$.
(Note that it is possible that no clock rings because $t_1>T$).
Now my question is: What is the probability that two or more clocks ring. Equivalently it would be enough to know the probability that no clocks ring or that only one clock rings.
I hope somebody can help me.

Answer 1

The number $Y$ of clocks that ring by time $T$ has Poisson$(\lambda T)$ distribution, so $P(Y=0)=e^{-\lambda T}$ and $P(Y=1)=\lambda Te^{-\lambda T}$.

See https://en.wikipedia.org/wiki/Poisson_point_process#Interpreted_as_a_point_process_on_the_real_line