The following threads are related to my question, but since I have minimal confidence around probability & math, I'm not sure how to interpret some of this and whether the answers apply to my situation (I think I'm dealing with a binomial process and most of the relevant material I've found describes Poisson processes):
- What is the probability of an event happening in some interval given probability of it in x interval?
- Probability of an event that has happened, to have happened in a specific time range?
- Calculate probability of an event occurring exactly once in an arbitrarily selected year, given mean annual occurrences over a century
Suppose I take "samples" of the environment in hopes of detecting something. I have a model that tells me the probability an event will occur at least once sometime between 7:00am and 8:00am. Suppose this probability is 0.25, and is based on an hourly weather prediction — so my predictive model is only good to the hour. However, I can only take samples in scheduled chunks of one minute. Assuming perfect detection, what's my probability of detecting the event of interest if I sample any SINGLE random minute out of the hour? What is my probability of detecting this event if I can randomly sample 2,3,4,5… or any N <= 60 minutes out of this hour?
I was thinking it would make sense for my problem to spread this 0.25 probability uniformly between the 60 minutes since I only have hourly resolution. So if I can only sample 1 minute out of the hour, probability of the event occurring at least once in THAT minute is 0.25*1/60. If I can sample 2 minutes out of the hour, total probability of the event occurring at least once during those minutes is 0.25*2/60. Three minutes would be 0.25*3/60 and so on.
Alternatively, I know it might be more conventional to say: 1-((1-0.25/60)^N) for calculating the probability of this event happening at least once across N sampled minutes, but this feels less intuitive for me within the context of a disconnect between the time scales.
Perhaps either approach is fine depending on the system assumptions. Regardless, is it okay to divvy up that 0.25 hourly probability among the 60 minutes the way I have done, or does this violate something basic in probability theory?
(If it helps provide context, the events I'm trying to detect are bird songs! I don't care about rates (Poisson), I just care about presence or absence of singing in a given sampled minute.)
Best Answer
Well, you should care about rates and Poisson processes! If the bird songs are independent of each other and of time, then you can assume a Poisson process with a stable rate of, say, $\lambda$ songs per hour. The Poisson distribution tells you that the probability of no song during one hour is $$ P(X=0)={\lambda^0 e^{-\lambda}\over 0!}=e^{-\lambda}. $$ If this probability is equal to .25 then you can deduce the rate $\lambda$ as follows: $$ e^{-\lambda}=1/4 \rightarrow \lambda=-\ln(1/4)=\ln4\approx 1.38629 \text{ song per hour.} $$ If you are interested in what happens over $x$ minutes, then the expected number of songs will be $x\ln4/60$ and the probability of detecting (i.e., at least one song over $t$ minutes) will be $$ P(X\geq1)=1-P(X=0)=1-{(t\ln4/60)^0 e^{-t(\ln4)/60}\over 0!}=e^{-t(\ln4)/60}. $$ For a 1-minute interval, this is $$ 1-e^{-(\ln4)/60}\approx 0.02284. $$ and for a 90-minute interval $$ 1-e^{-90(\ln4)/60}\approx 0.875. $$