Elevator Question: Odd of Selecting a Particular Floor?

Question

I just got on an elevator in a builing with fourteen floors. There we seven of us on the elevator, and when everyone had finished making their floor selections, everyone had selected a different even numbered floor. All even numbers from 1-14 were selected, and since there we only 7 of us, every individual chose a different number. This stuck me as a rare occurance, so I'm wondering - what are the odds of this happening?

Answer 1

First, I assume nobody is going to press 1... that leaves 13 floors. (If your building has a ground floor, change 13 to 14 below. If your building skips the 13th floor as many do, change 13 to 12.)

Each person has 13 choices for picking a floor. We also assume the events are independent so if someone picks 3, I might also pick from any of the 13 floors and pick 3 also. (This is where the person above made a mistake).

For your outcome to happen, the first person must pick an even number floor 2,4,6,8,10,12,14 (7/13)

Then the second person must pick a different even number floor (6/13).

The third person must pick a different even floor still (5/13), etc.

So we have 7/13 x 6/13 x 5/13 x 4/13 x 3/13 x 2/13 x 1/13
Written equivalently this is 7! / 13^7 or 5040 / 62748517

For a building with floors {1, 2, 3, ...., 12, 13, 14} this is about 1 in 12450.

P.S.
For a building with floors {G, 1, 2, 3, ..., 12, 13, 14} this is
7! / 14^7 or 5040 / 105413504
or 1 in 20915

For a building with no ground floor and no 13th floor {1, 2, 3, ... 12, 14} this is
7! / 12^7 or 5040 / 35831808
or 1 in 7109

So whatever the case, this is a pretty rare occurrence but not beyond the realm of possibility.

Answer 2

First, I assume those who go to the first floor does not need to go into the elevator. So, there are actually 13 choices.

You have 7 people. Each one can make an independent choices from the 13 numbers. How many ways can they make the choices? that is 13^7=62748517.

Now how many ways are there for 7 people to select among 7 numbers 2, 4, 6, 8, 10, 12, 14, and each select a different one? That is 7! = 7 *6 * 5 * 4 * 3 * 2 * 1 = 5040.

So the probability is 7! / 13^7 = 0.00008032

Answer 3

Realistically, the floor selections aren't random -- at least not in any useful way. For instance, if it's an apartment building, we'd have to know how many people live on floor 6 compared to floor 10, for instance.

But if we pretend that the floor selections are truly random, we could imagine that:

Person1, odds of selecting an even floor: 7 / 14.
P2 odds of selecting one of the remaining even floors: 6 / 13
P3: 5 / 12
P4: 4 / 11
P5: 3 / 10
P6: 2 / 9
P7: 1 / 8

Combined Probability: ( 7*6*5*4*3*2*1 ) / (14*13*12*11*10*9*8 )
≈ 0.029%

Answer 4

Each person has to pick a particular floor out of 14 possible. So you could start by asking, for example, "What are the odds of Mary selecting the 2nd floor?" The answer, of course is 1/14 or about 7.14%. Since the odds are the same for each person, and there are seven people, we can raise the odds for one person selecting a particular floor to the seventh power. This would be 1/14^7 or 1/105413504.

Answer 5

Maybe you were in one of those posh buildings where every flat is split level, and the patron's main entrances are on even numbers, and the servant's entrances on the odd numbers.
Such buildings do exist, in Hong Kong, for example....
The fact that everyone goes to a different flat is not so unusual, after all, an apartment building is not a Kibbutz.,...

Answer 6

More information is needed to solve this problem. Because of supersition, it is common for buildings not to contain a 13th floor. The probability will vary based on the absence or presence of a 13th floor.

Answer 7

100% in your case