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Gamow's Elevator (Posted on 2020-07-26) Difficulty: 3 of 5
George Gamow and Marvin Stern occupied offices on the second and sixth floors of a seven-story building, and noted that when either took the elevator to the other's floor, it was going the wrong way. It's apparent why: there were ten segments of the elevator's 12-segment cycle (6 going up and 6 going down in a continuous cycle) where the first elevator arrival would be going the wrong way and only two segments where it would be going the desired direction the next time it passed the boarding floor.

But what if a second elevator were placed in the building. What would the probability be that the next elevator to arrive would be going the wrong way? Ignore stops along the way, as they do not affect the distance that need be traveled and probably have more of them for longer trips. The two elevators move independently of each other.

Gamow himself did not get the correct answer for the two-elevator case, but the correct answer was found by Donald Knuth.

  Submitted by Charlie    
Rating: 1.5000 (2 votes)
Solution: (Hide)
If both elevators (say for Gamow, on the second floor) were above the second floor, with probability (10/12)^2, the next elevator would certainly arrive going down (the wrong direction for Gamow to board).

But the other way (mutually exclusive) would be for one elevator going down between the 4th and 2nd floors while the other was making a round trip from the second to the first floor and back. That would be the case (1/6)^2 of the time if we considered elevator 1 to be the one above the second floor and elevator 2 below, so we have to double it, but then we have to halve it as the probability is 1/2 that the one above the second floor is farther in its 2-story journey than the one below the second floor, effectively countering the doubling.

That makes the overall probability the next elevator is going the undesired way (10/12)^2 + (1/6)^2 = 13/18 of the time.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
ccoaboxnovel2020-07-30 22:22:59
soln - w/ logic and computerSteven Lord2020-07-28 21:45:44
re(4): what I have so far... -- HintSteven Lord2020-07-28 07:15:12
re(3): what I have so far... -- HintCharlie2020-07-28 07:03:03
re(2): what I have so far... -- HintSteven Lord2020-07-27 22:53:52
Hints/Tipsre: what I have so far... -- HintCharlie2020-07-27 21:51:43
what I have so far...Steven Lord2020-07-27 19:03:50
clarificationsCharlie2020-07-27 07:03:52
The extreme casebroll2020-07-27 05:09:53
I understand the problem?Steven Lord2020-07-27 01:05:08
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