January 9, 2010

The Probability and Politics of the Toilet Seat

I live in one of the largest rooms on campus: a 5-person senior suite in the Fahey/McLane dorm. Our room complex has its own restroom, which is used exclusively by roommates and our guests.

One of our number, who had appearently taken one too many government classes, decided that restroom guidelines were needed to ensure its orderly use. This is the blitz he sent.
>Subject: leaving the toilet seat up
>To: Suite Members
>From: Roommate #1

in the suite is an appropriate, acceptable, and manly thing to do. 9 times out of ten, regular use of this toilet calls for the seat to be raised. as such, it is a social courtesy within this limited context to leave the seat in its most convenient position for the restroom's next probable occupant. these cold, hard statistics are not even counting booting, which, if last night was any indicator, is to be applauded as a rising trend on the zeitgeist. clearly, when social dynamics indicate that women are present, an exception and reversion to traditional social convention will be in order. otherwise, however, let us embrace the revolution
One of our resident math majors crunched the numbers and sent out this toilet-libertarian rebuttal that is as comprehensive as it is crushing.
>Subject: Re: leaving the toilet seat up
>To: Suite Members
>From: Roommate #2

Isn't it more efficient to just leave it the way you leave it?

Let's run some numbers:
Suppose that we stand 80% of the time and sit 20%. If we were to always put the toilet seat up, and we assumed that we were perfect and would never fail to put something back into it's proper position after using it *cough* then we would not need to manipulate the toilet seat 80% of the time and but would have to manipulate it twice the other 20%.

If we were to just leave it the way we leave it, then on times we need to stand, we would find the seat up with probability .8 and therefore would not need to manipulate it and would only need to manipulate the seat once with with probability .2. In the other case, when we need to sit down, we would need to manipulate the seat once with probability .8 and not at all with probability .2. In sum we would need to manipulate the seat once with probability .8*.2 +.2*.8 or .32.

So if we follow [the first author]'s proposal, in 100 bathroom visits, we would expect to manipulate the toilet seat .2*2*100=40 times, but with my proposal we would only have to manipulate it .32*100=32 times. So we could expect to save ourselves from moving the toilet seat 8 times per 100 bathroom visit by following my procedure over [the first author]'s.It can be proven that my proposal is more efficient no matter what percentages we assume for a standing/sitting down ratio, but I'll leave that as an exercise for the reader.

Of course, this analysis does not take into account other factors such as concentration of the sitting action towards morning(or afternoon on Tuesday's Thursday's Saturday's or Sunday's) hours or a few hours after Gusanoz is ordered, where the the efficiency of my proposal increases, Nor does it take into account that making a rule that toilet seats are to always be up would force the assumption unto us that toilet seat will always be up despite human error. This assumption could cause disastrous consequences and it would be better for us to be vigilant upon entering the bathroom.

Well done boys. Well done.

1 comment:

  1. Solution for the "exercise for the reader":
    The expected number of times the toilet seat would have to be manipulated in the first scenario per one hundred uses can be expressed as
    "2*x*100 = 200x", where x equals the percentage of uses that involve sitting down and where the toilet seat is always returned to the upright position.

    The forumula for the number of manipulations per 100 uses in scenario 2 can be expressed

    Therefore we can see that for all positive percentage values of x, 200x > 200x-100x^2, and therefore option 2 will always be more efficient than option 1 by 100x^2 manipulations.