Don*t you have an experience as mentioned in the next paragraph?

Waiting at the 4th floor elevator hall of the 5-storied building, to go down, elevators one after another are to go up. You cannot catch one. You start thinking what a bad luck.

5-storied building case might be tolerable, but if you are on the 50th floor of a 60-storied building, you notice almost all the elevators operating upwards. How come these things can happen?

At first, let us consider as next, using the simulation on the right. Imagine, one observes on the 4th floor. Let us investigate the times of:

(1) elevator being on the 4th floor and above

(2) elevator being on the 3rd floor and below

(3) the trials or observations

From these elements, we can find the probability of an elevator coming up, when the observer comes to the elevator hall, at random timings.

[assumptions]
(1) The observation shall see, on the 4th floor elevator hall of a 5-storied building, at random timings, whether the elevator is on the 4th floor and above, or on the 3rd floor and below.

(2) There should not be any bias among the available floors. We don*t consider a case where there is a supermarket on the 2nd floor while the 3rd floor and above are residential.

On these assumptions, we need to find the probability (ratio) of:

(1) elevator being on the 4th floor and above

(2) elevator being on the 3rd floor and below

How do you, yourself consider?

Please examine in the following manner. Assuming the observer is on the 4th floor, the sample space is the floors 1 through to 5. In order for the observer on the 4th floor, to come across with an elevator going down, an elevator should be in the area colored in pink. We call this event as A. Then the dimension of sample space and of event A can be determined respectively:
-dimension of the sample space = (width h) x (floor height v) x 5

-dimension of event A = (width h) x (floor height v) x 2

Therefore, the probability p we need to find is In fact, you can confirm this using the simulation.

This elevator paradox first appeared in the book "Puzzle-Math" written by physicist George Gamov and his friend Marvin Stern.
Later, computer scientist Donald Knuth, famous for the TeX - of Stanford University, in his paper "The Gamov-Stern elevator problem", indicated that the probability of catching an elevator either going-up or going-down would converge to 1/2 at all floors, in proportion to the increase of the numbers of elevators. If interested, please refer to "The Art of computer Program - the Mathematics Building of California Institute of Technology Section 2.2.5 Donald Knuth. In the next chapter, we do its simulation.

What about an experience as the following?  Often, after the departure of an express train, a local train would depart as if catching up the express. In these cases, I feel like I cannot ever take express trains, unless waiting a long time.

We now assume the timetable as illustrated on the left. Here, let*s find the probability of catching an express train, in a catch-as-catch-can manner, at random timing between 1:00 P.M. and 2:00 P.M.

Let us first, calculate the time to take an express train.

If it were 13:02, next available is 13:10 which is a local. In order to catch an express train, one must arrive on such timing as shown colored in red in Fig.1.

Now, let sample space be 60 minutes while event of taking an express train be A.

sample space = 60 minutes

event of taking an express train=6 minutes (sum of red part in Fig.1)

Therefore probability p of getting an express train is: Exercise 1 At a certain bus stop, between 9:00 A.M. and 10:00 A.M., the departure times are as follows. 9:05 9:10 9:25 9:45 Find a probability p of a person not knowing the timetable, arriving between 9 A.M. and 10:00 A.M., to come across with a bus within 5 minutes.

There are many interesting problems such as how many elevators could be good enough for realizing more than 0.5 probability of catching an elevator on the 4th floor to go down. Among the events close to our daily life, we can find quite many occasions implying the notion of probability. 