I am trying to find the probability that two Poisson Distributions will happen consecutively.
The context is that, given two datasets that describe the number of sightings of bears from randomly selected expeditions in a forest, respectively during daytime and nightime.
In each file there are two columns describing both number of sightings and according time spent in the forest during the expedition.
$ sightings_{i} = [2, 0, 4, 1 ,\ ... ], hours_{i} =[3.7, 6.1, 2, 13.2,\ ...], $
$\lambda = \frac{\sum_{i} sightings_{i}*hours_{i}}{\sum_{i}hours_{i}} $
Estimate the probability that at least one sighting $P(X\geq 1)$ happens during 12 hours, 9 of which at night and 3 at day time. This is the part that throws me off.
My logic tells me this could be simply solved by :
$P(X_{night} \geq 1\ or\ X_{day} \geq 1) = P(X_{night} \geq 1)\ +\ P(X_{day}\geq 1)\ -\ P(X_{night} \geq 1| X_{day} \geq 1)P(X_{day} \geq 1)$
$=P(X_{night} \geq 1)\ +\ P(X_{day}\geq 1)\ - P(X_{night} \geq 1)P(X_{day} \geq 1)$
But I don't know what to do of the 9 hours at night and 3 at day ($\lambda_{night}*9, \lambda_{day}*3\ ?$), and in general cannot really grasp what is the intuition (i.e. can we assume independence ?) behind all of it.
I would appreciate your help, thank you :)