# How big is the chance that i will find the elevator ready for me?

Asked by

rebbel (

28487)
March 26th, 2010

I know zilch about mathematics and i think i should need to, to solve my question.

In my apartment building we have an elevator which i use to go up and down from/to my house.

Every now and then when i go to the elevator i’ll find it standing ready on my floor (the 8th), and then i wonder how big is the chance that it *is* standing ready there for me.

It’s a nine story building with 6 houses on each floor.

We also have a stairs (if that matters to solve the equation).

I realise this is not the most important ‘problem’ that needs a solution but maybe you are keen to help me?

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## 26 Answers

Without knowing the distribution of occupiers, their ages, number of children, working hours, the time of day etc., it would be just a guess.

@DarkScribe Not sure if you’re serious, but i was actually thinking this morning whether i should tell that their are about 40 percent elderly people living in this building.

Most of them (don’t know exact figures) living on the bottom four floors.

**EDIT** I would like to add to my question, that i’m also interested if you could ‘just’ give me guesstimations (see @DarkScribe‘s reply above).

@DarkScribe

But then again, all probability assessment is based on not having enough data. ;) Isn’t probability assessment the art of guessing optimally?

If there are nine floors where it can stop, and we know nothing of which floors gets most elevator traffic, I think we’ll need to assign each floor an equal probability. Until we have enough information to sophisticate that distribution.

Assuming no “travel time” between floors, then at any moment the elevator is on one of 9 floors (including the ground floor, and assuming there is no basement). So you have a one in nine chance of finding the elevator at your floor at any time you hit the button to summon it.

Of course, that doesn’t take into account actual travel time (which can’t be ignored) or people holding the elevator at one floor or another, or the chance that more people make more trips from all of the other floors, and “only you” travel to and from the eighth floor.

So what happened to you the other day may be a once-in-a-lifetime event. Hope you enjoyed it.

My guess:

You have 9 floors, so that means that there’s a 1/9 chance that it will be waiting at your floor.

If we assume that it takes one second for you to press the button and for the doors to open, and that it takes about 3 seconds for the cab to get from one floor to the next, then my guess is about ½7.

I.e., the chance that it’ll be waiting at your floor when you get there is about 1 in 27.

Other factors, such as the elevator-using traffic on other floors compared to yours, would change this ratio.

@rebbel *DarkScribe Not sure if you’re serious,*

Families with kids will have the elevator on their floor more often than singles or couples. People who work will have distinct patterns with no usage while they are working. Elderly people are likely to use the elevator during the non school/work hours.

@Brian1946

I don’t understand. How did you decrease that probability from 1/9?

Do you mean you’re taking into account the probability that it’s moving between floors at the moment? (That’s clever, I didn’t think of that.)

But for that you also need to know how long it stays on one floor. If for example it takes it three seconds to move from floor to floor, and it stays on each floor for twelve seconds, the probability at a given moment that it’s not moving is four times as large as the probability that it’s moving, right?

So with those (arbitrary) numbers you would divide the probability per floor by five and multiply by four, so you would get 1/11,25 or 0,08889.

(Still not taking the demographics into account.)

@All and @DarkScribe I guess there are just too many unknown factors to make it possible to find an answer.

After all, we don’t know (yet) the time it takes the elevator to travel from ninth to first, how many people take (like i do every once in a while) the stairs to go down, let alone the number of kids who, just for fun (like i did when *i* was a young boy) push all the buttons just before they exit the elevator.

@rebbel

So bring a stopwatch the next time you use the elevator. :D

I still think you should be able to at least get a provisional assessment of the odds even without all the data you can get…

@Fyrius

“Do you mean you’re taking into account the probability that it’s moving between floors at the moment? (That’s clever, I didn’t think of that.)”

You’re right- that’s what I mean and thanks.

“But for that you also need to know how long it stays on one floor. If for example it takes it three seconds to move from floor to floor, and it stays on each floor for twelve seconds, the probability at a given moment that it’s not moving is four times as large as the probability that it’s moving, right?”

You’re right about that- the average amount of time that the elevator doors stay open while other people are getting in or getting out would be another factor.

All right, here’s a feasible, sure-fire way to find out the most accurate chances possible.

(It’s probably too much work to actually do, and you probably don’t really want to know that badly, but it’s fun to design experiments.)

Screw calculating a thousand variables, we’re going to collect direct observations instead.

You need a way to measure how much time of the day the elevator is at your floor. Ideally you should have a device do it for you, so you can record times for entire days during a few months or so. If you know what percentage of the time it’s on your floor, you know the probability that it’s there at any given moment. It would be the same number.

Your device will need a way to tell whether or not the elevator is at your floor. I think you could do that with a short range wireless receiver taped to the frame of the elevator door or something, and a wireless emitter inside the elevator. You can probably get suitable hardware from a cheap wireless keyboard, for example. (Hide them well, because if people see them and wonder what they are it will affect how long they stay in the elevator and mess up your data.)

And then you link the receiver to your computer, and write a program that keeps track of what portion of the time the receiver makes contact with the emitter. And keep it running all the time.

@Fyrius No, i don’t want to know it *that* badly, you guessed right.

Thanks for your input, though.

I like your way of thinking (”*Hide them well, because if people see them and wonder what they are it will affect how long they stay in the elevator and mess up your data.)*”), the same thoughts that go around in my head.

If you have a building engineer, they may provide some insight. Newer buildings have an elevator system where you know exactly where each car is at any given time by looking at the panel. I think this may be required by code for fire safety where I am, not sure. Anyway, they may log this data. Another thing to consider is how many elevator cars you have in the building. Usually, an apartment building has at least two – one for moving bulky items and one for regular every day use – but both are operational 24/7.

Shame, it would have been cool as heck to be able to see the probability vary by the time of day, day of the week and season of the year. ;)

@lilikoi

Wow, that would be even better. If you can get your engineer’s data you don’t even have to do the work yourself.

It’s probably standing there open because the last occupant was delivered to your floor. I would just think of it as random. Trying to make it a math problem of it would give me a headache.

Those who say that it’s a 1/9 probability are basically saying that it should happen once every 9 times. But I highly doubt that that is the case. In a dynamic environment, you would be hard pressed to finger a number down for this subject. . . don’t lose sleep over this.

@rebbel I think it’s just a good karma day when it’s waiting for you.

I’m not good at math

I am amazed at all the wrong answers!

People will use the elevator to go either up or down. Assuming there isn’t a lot of visiting other people on different floors, there is a 1 in 18 chance it will be on your floor, since everyone will go down to the 1st floor to leave.

There is an interesting thing about elevators that most people don’t know and would never guess.

They have floor settings. They have commands in place instructing them what floor to remain on during periods of inactivity…usually dependent on where the elevator last stopped.

And there are those elevators which always return to the first floor after a period of inactivity.

And some elevators merely wait on the last floor they were occupied until called into service.

@davidbetterman

Now *there* is something i indeed didn’t know, and quite interesting. Thanks.

@ All

Thanks for your replies!

@rebbel

You’re welcome. :) This is a fun thread.

Mind if we go on for a bit more?

@thriftymaid

Well, yes, it is random, but that doesn’t mean you can’t tell the probability. Dice are random and their faces all have a 1/6 probability.

@filmfann

I see! The first floor would get much more traffic. Clever.

But if we imagine there’s no elevator traffic except from people going from their apartments to outside or the other way around, then the people who live on the first floor would never use the elevator. So then every other floor would have a 1/16 chance, but the first floor would have a 8/16 (½) chance, because every trip to and from the upper eight floors would have one stop at their home floor and one stop at the first floor. Right?

For those who still like to do calculations (assuming my freshly found data is valuable), here are the times i timed:

-From the moment the car comes to my calling and stops at my floor, it takes 15 seconds from stopping – opening-the-door – departing.

-From the eight floor to the last (0th) floor it takes 27 seconds.

Have fun!

All of your calculations to get the probability of the elevator being at that floor don’t mean squat when @rebbel is not at that elevator door which is what the question asks for anyway.

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