# The unlikely day

Today I was "late" to work (later than usual - I try to get in before 8:30, but I arrived and it was almost 9) because school started at UW and, besides being normally late to catch my bus, the first bus that went through my stop was so full that the bus driver didn't even stop. And then the bus that I eventually caught stopped way more than usual. On my way in I saw an probabilistically unlikely event:

The building I work on has 11 floors (American way of counting, meaning that the ground floor is counted in these 11). 7 people got into the elevator and pressed: 4, 5, 6, 7, 8, 10, 11. 7 different floors for 7 people. How likely is that? Let statistics answer the question:

Model 1: all floors have the same likelihood to be selected (1/10):

10/10*9/10*8/10*7/10*6/10*5/10*4/10 = 6.0%

Model 2: the first two floors have half the chance of being selected (because many people that work on those floors walk up the stairs instead of waiting for the elevator)

18/18*16/18*14/18*12/18*10/18*8/18*6/18 = 3.8%

Model 3: the same as model 2 but taking note of the fact that nobody chose floors 2 or 3 (which I think is a little bit of a stretch to consider, but it makes the number pretty small, which enhances the point)

16/18*14/18*12/18*10/18*8/18*6/18*4/18 = 0.8%

So statistics says it's not very likely, but it doesn't seem as rare as my intuition says (maybe except for model 3, but that was added "arbitrarily"). Maybe it's because I have this policy that if somebody presses the floor just above or below me I'll just use the stairs for the difference. However, this time I was the first one to enter the elevator, so I wasn't able to make that decision.

Anyway, I probably should get back to work here. To add to the unlikeliness of today, I'm doing a third blog post is less than 24 hours. Quite odd indeed.