Travis Gockel
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 Oct 1 awarded Supporter Oct 1 comment The Probability of $k$ events in $n$ systems within some time period Yes -- $t_{\Delta}$ is basically constant (in reality, it is dependent on the number of lights that are on, but it varies so little it probably doesn't matter). And if I understand your second statement, then yes to that as well. I edited the question with a rephrasing of the problem at the start...hopefully it will be better than my first attempt. Oct 1 revised The Probability of $k$ events in $n$ systems within some time period Restatement of problem for clarification. Sep 30 comment The Probability of $k$ events in $n$ systems within some time period All lights begin as off. Sep 30 comment The Probability of $k$ events in $n$ systems within some time period It might be easier to explain this way: I have $n$ lights that turn on with an average of $s$ seconds between turning on and they stay on for $t_{\Delta}$ before turning off (so exponential distribution with $\lambda=1/s$). A light cannot turn on again until it is turned off, but can turn on and off multiple times. What I would like to know is the likelihood that $k$ lights will be on at the same time in the window I care about ($t_{*}$). Sep 30 comment The Probability of $k$ events in $n$ systems within some time period I'm not sure what more you are asking for. The only assumption is that my $n$ systems are properly modeled by $\lambda e^{-\lambda t}$. What I'm looking for is the likelihood that, given $n$ systems, what is the likelihood that $k$ events will happen within some time period of each other. Sep 30 asked The Probability of $k$ events in $n$ systems within some time period Nov 29 accepted Probability Two Birthdays are within M Days Nov 29 awarded Scholar Nov 29 revised Probability Two Birthdays are within M Days added 8 characters in body Nov 29 revised Probability Two Birthdays are within M Days deleted 2 characters in body Nov 29 awarded Editor Nov 29 revised Probability Two Birthdays are within M Days added 243 characters in body Nov 29 awarded Student Nov 29 asked Probability Two Birthdays are within M Days Nov 29 awarded Autobiographer