Episode 410 - "Chinese Box"

In this episode a paranoid man storms into the FBI building and holds David hostage in an elevator, leaving Charlie and the FBI to scramble to find a way to free their friend and colleague.

Early on in the episode, Charlie and his dad are waiting in the FBI bullpen for the elevator. Alan comments, "You ever notice how the first elevator is always going the wrong way?" Of course, Charlie cannot resist the opportunity to talk math with his father. "Actually, the Elevator Paradox accounts for that…any one elevator spends most of its time in the larger section of the building, and is more likely to come from that direction when you hit the call button. If you stood here for several hours…". Right then, the elevator arrives, and Alan cuts Charlie off by saying "Saved by the bell!"

The Elevator Paradox was first noted by George Gamow and Moritz Stern, two physicists with offices located on the second and sixths floors, respectively, of a seven-story building. Both men noticed that, more often than not, arriving elevators were traveling in a direction opposite to the direction they wanted to go. At first glance, one might deduce that, since an elevator spends more time in the larger section of the building, it is therefore more likely to arrive from that direction as it approaches a location near the top or bottom of the building.

However, the notion that people near the top or bottom of a building are at a disadvantage is based on flawed reasoning. This paradox can be explored with students using fairly straightforward probability theory. Consider for example, that Gamow was located on the second floor of this 7-story office building containing a single, really slow and deliberate elevator. By deliberate, let’s assume that the elevator moves in one direction, stopping at each and every floor until it reaches either the top or bottom floor at which time it reverses direction. Finally, assume that the elevator travels at a very slow and constant speed, taking exactly 1 minute to move from one floor to the next (this includes the time for the doors to open and let passengers on and off the elevator). You could use this information to have students create an elevator schedule similar to the one shown below.

Floor	Arrival Time – Going Up	Arrival Time – Going Down
1	8:00, 8:12, …	n/a
2	8:01 8:13, …	8:11 8:23, …
3	8:02 8:14, …	8:10 8:22, …
4	8:03 8:15, …	8:09 8:21, …
5	8:04 8:16, …	8:08 8:20, …
6	8:05 8:17, …	8:07 8:19, …
7	n/a	8:06 8:18, …

Now, consider someone located on the second floor wishing to go in the upward direction. During the course of 12 minutes, the elevator makes 12 stops. For 6 of these stops, the elevator is traveling up and for 6 of these stops, the elevator is traveling down. Of these 12 stops, 2 of them (8:01 and 8:13) are at the second floor traveling in the desired direction. Expressing this using a probability ratio, we get:

Therefore, the chances that the elevator’s next stop is your floor and your direction is ¹/₆.

But here’s where the paradox ends. It doesn’t matter what floor you are on – the probability that the elevator’s next stop is your floor and direction is ¹/₆ regardless. In fact, if you were to observe every elevator’s arrival for an extended period of time, you would notice that the number of arrivals traveling in the downward direction would be equal to the number of arrivals traveling in the upward direction at any floor.

As a former electrical engineer for the Otis Elevator, I can’t help but be amused at the irony of writing this particular blog. I also must interject a dose of reality to this situation. For example, most elevators are programmed to anticipate the changing demands of a typical day. In the morning, elevators typically idle at the lobby since most passengers are arriving and traveling upwards. This will certainly have an impact on the elevator paradox. In fact, a waiting passenger located at the top of an office building will feel this paradox even more acutely during the morning hours. There are other factors that complicate things such as full elevators skipping stops or lopsided demand for downward traveling elevators at the end of the workday.

Click here to see an applet that simulates a situation with 6 elevators in a one-hundred story building. In this applet, you will see three images. The first image shows the actual movement of each elevator. The second image shows a graph of the average wait time for elevators located at the 10th, 20th, …, 90th floors. The results of this graph can be extrapolated to the wait times for all 100 floors. Finally, the last image shows the probability that the arriving elevator comes from above rather than below. This simulation is based on four basic rules.

1. The floor in which demand occurs is random.
2. The destination floor is also chosen at random.
3. The elevator that responds to a call will be the elevator closest to the floor in which the call originates.
4. Simultaneous demand from two different floors will not occur.

Click the start button to initiate the applet and run it for as many trials as desired. You also have the option of stopping the simulation and resetting it at any time.

You may also want to extend this concept to other applications. For example, you may want to obtain a copy of a local train schedule and ask students to calculate some probabilities. Below is the train schedule that I would use if I were to travel to Boston.

Here’s one question you might ask: "If I were to randomly show up at the Norwood Central stop between the hours of 6:00 a.m. and 9:00 a.m., what is the probability that my wait time would be 10 minutes or less?" The sample space for this experiment would be 3 hours or 180 minutes. Since there are 7 trains during that time span, there are a total of 7 • 10 = 70 minutes that are favorable. Therefore, the probability of waiting 10 or fewer minutes is:

This application can be further analyzed as a binomial experiment. Ask the question, "If I were to arrive randomly at the Norwood Central stop for an entire week between 6:00 a.m. and 9:00 a.m, what is the probability that I would have to wait 10 minutes or less exactly 3 times? 4 or fewer times? 4 or 5 times? etc…" The first part of this question can be answered by evaluating the expression:

Using graphing calculator technology, this result can be found using the binompdf command as shown here.

Steve Ouellette, Westwood High School, Westwood, MA