A building has $10$ stories ($i=1, 2, \ldots, 10$, where $i=0$ is the ground floor). Each of those stories have $4$ apartments. Inside each apartment there is a married couple. The ground floor, $i=0$, has no apartments in it.
- The probability that a given couple have no children is of $50\%$.
- The probability that a given couple have only $1$ child is of $30\%$.
- The probability that a given couple have $2$ children is of $20\%$.
It is also known that:
- All children are at least ten years old.
- All residents have the same probability, $p$, of taking the elevator at a certain time (either down to the street, or up to their apartment)
In a given instant of time ($t^*$), there are $80$ residents inside the building. There is no information about how they are distributed, but we do know there is at least one couple in their apartment, per story.
Exactly at that moment $t^*%$, the milkwoman arrives at the building. What are the chances that she will (fortunately!) find the elevator at story $i=0$?
EDIT 1 Since this puzzle has a downvote, there is something I obviously did wrong. Therefore I will try to make one thing clearer.
It is regarding the probability $p$: this is the probability that a given resident (note that we are not considering random people like pizza delivery guys or plumbers, but only residents), takes the elevator:
- If a resident is at home, there is a chance that he or she takes the elevator (down to street) equal to $p$.
- If a resident is outside the building, there is a chance that he or she takes the elevator (back to apartment) equal to $p$.
- All residents, inside or outside the building, has an equal chance of taking the elevator.
- A resident can either take the elevator from the apartment to the street, or from the street to the apartment. These are the only two possibilities considered.