# The Building, the Elevator and the Milkwoman

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.
• 100% chance that it's on the ground floor, because the elevator is broken :)
– user16469
May 27, 2016 at 20:12
• Clever! But not the answer, unfortunately. The elevator does work! May 27, 2016 at 20:13
• (1) I believe I understand the configuration you're describing, but, if a building has ten stories above the ground floor, then that building has eleven stories.   (2) I know of a building in which the elevator automatically returns to the ground floor when it is not in use.  I've heard of buildings where the elevator will idle at a point halfway up the building (i.e., the 5th story), to minimize the average wait time when it is called.  Unless you've got some crazy trick up your sleeve, it will be impossible to solve this question without a description of how the elevator behaves. May 28, 2016 at 4:58
• Please do not respond in comments; edit your question to make it clearer and more complete. May 28, 2016 at 4:58
• Not a tricky question, Peregrine. The elevator behaves normally, it doesn't return to the ground floor automatically. Last resident to use it will define the floor it is in. May 28, 2016 at 10:39

$\frac7{27}$

If we also know that all the children are residents then there are

$108$ residents
($10\times4=40$ couples and their $10\times4(0.3+0.2\times2)=28$ children)

There are $80$ residents at home

and $108-80=28$ not at home

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)

So (ignoring other elevator users such as guests and those making deliveries)
...all residents were equally likely to have been the last to use the elevator.
Thus there is a $\frac{28}{108}=\frac7{27}$ chance that the elevator was last used by one of the residents going down to the street leaving it at the ground floor for the milkwoman.

• But the last resident to use it might have just arrived home, leaving it in his floor May 27, 2016 at 22:02
• @JoseLopez there is a 20/27 chance that the last resident to use it might have just arrived home, so I think this answer looks good. May 27, 2016 at 22:29
• Brilliance. I guess this was not a hard puzzle to solve at all. Maybe that's the source of downvotes? Anyway, well done! May 28, 2016 at 16:02

2 possible scenarios:

• If it is the morning after the parents have gone to work, and the children have just gone to school, then the probability is "close to 1.00", that the elevator is at the ground floor when the milkman/lady arrives!

• Late at night the opposite situation would occur! The probability that the elevator is at the ground floor, would be "close to 0.00", since there are no apartments at the ground floor, and most have returned from work, school,(and other activities of daily life!)

• With 40 couples, 30% (12 couples) have 1 child, and 20% (8 couples) have 2 children. This adds up to 28 children. Given that each resident has the same probability $p$ of coming or leaving at any given moment, we would expect, at any given moment, to have about half of the people there and half of the people gone.
• Now consider the state just prior to $t^*$ - either we had 79 people in the building and one came home, or 81 people in the building and one left. Now we just have to figure out the probability of each scenario occurring, but unfortunately this is not possible without knowing about the prior state of the building. If the building starts off with more than 80 people home, then it's more probable that $t^*$ happened right after the 81st person left their apartment, but if the building starts off with fewer than 80 people home it's more probable that $t^*$ happened right after the 80th person came home. In the former case the probability of the elevator being at the ground floor is higher than 50%, but in the latter case the probability is less than 50%.