Toilet seat up - Toilet seat down. It is certainly important that one chooses a proper position applicable to your particular need per toilet visit.  When standing to pee the correct position is, of course, up.  One might suppose that is about all there is to say about toilet seat positions.  However, for many households, U.S. households in particular, this does not seem to be the case. Indeed, before posting this write-up I was surprised to find that so much has already been written within E2 on the subject of toilet seat management. And much of it with passion. (Example)

I amused myself for over a year thinking about the impacts of different toilet seat administration policies and how to measure them – doing calculations in my head, considering ratios of Standing events to Sitting events, and I slowly began to understand some of the specific differences in the basic policies that know to be administered most often. Finally, I decided to perform a probabilistic analysis to compare two common policies that are both simple to understand and administer. As the following analysis will demonstrate, following one policy is materially different from the other.

My analysis compares and contrasts two very common toilet seat policies applicable to households where combinations of Standing events and Sitting events occur. I also tried to avoid a sexist point of view as much as I could (though the sexes can be often inferred). Rather, I produce actual mathematical results applicable to any household where members Stand or Sit to pee and Sit to not pee.

Note: I understand that other policies exist.  For example, a policy that requires all members to sit to pee.  Some have argued superior performance of such a policy, but that could be strongly debated.  Without taking too much time now for discussions of other policies and practices and why I chose to ignore them I'll immediately move forward with one particular analysis of two policies that I have found to be very common.

A. I analyzed specific differences between the following two policies:

1) “Sitter-Centric" – Households that require members who perform Standing events to move the seat to the up position before taking care of business and then moved to the lowered position when they complete their toilet visit.   Proper administration of this policy should find the toilet seat always in the down position.  This policy will require a household member to touch the seat twice for all Standing events; once to raise the seat and once to return the seat to the lowered position and to never touch the seat for Sitting events.
2) “Shared Responsibility” – Household members are required to raise the seat before Standing events, if and only if, those events have directly followed a Sitting event, and then allows the seat to remain in the up position. In this scenario household members will be required to lower the seat before Sitting events, if and only if, those events directly follow a Standing event. In general, this policy allows all household members then leave the seat at the conclusion of their visit in the same position they selected at the beginning of their visit.

In my opinion, a significant driving force behind most toilet seat administration policies is to eliminate or reduce the sitting in pee.  Most of us probably don't want to sit in our pee and even less of us care to sit in (or clean up) someone else's pee; so to start with, all successful toilet administration policies should have all members performing Standing events to first put the seat in the up position. Additionally, I noticed specific differences in how the two policies I examine require the toilet seat to be touched and considered that some people might value a policy that requires the fewest amount of seat touches.

B. My analysis should sufficiently answer the following questions:

1) Does one policy minimize the total number of times required to touch the toilet seat?
2) Does one policy provide more protection against finding pee on a seat when errors take place during Standing events?
As was possible when choosing which policies to analyze, I could delve into other questions.  However, I selected these questions because I felt they represent negative, but controllable, aspects of visiting a toilet. I don’t include the aesthetic beauty of always putting down the lid (and others offer tangible benefits that a lowered lid also reduces fumes, keeps objects falling into the toilet, etc.). Neither do I measure the embarrassment of a member hastily visiting the bathroom in the dark and promptly plunging into the toilet because the seat was left up before their Sitting event.  All of these incidental conserns and what if's are valid, but just not part of my analysis.

C. The mathematics of my analysis:

I use probabilistic modeling and therefore all values reported are actual expected probabilities over a large number of observations. I model the binary function of "I am at the toilet to perform a Standing event" versus "I am at this toilet to perform a Sitting event". The single parameter, p, of this function is the probability of a Standing event. A Sitting event is therefore the complement of the probability of p and is equal to (1-p). I later introduce and consider an error rate, E, and compare the merits of the two policies by testing the discovery rate, D, of errors against the modeled results of both policies. Both polices, when executed 100% successfully, will ensure that no one sits in pee; but perhaps one is more advantageous than the other given the real world observation of errors. Certainly, you will see how they differ.

D. Assumptions of my analysis:

(Therefore my conclusions may apply less, or not at all, to alternative household policies, or toilets without manually movable seats, etc.):
1) The household consists of members performing both Standing events and Sitting events. (This is not actually required, but certainly the entire comparison become mute if the household only has members performing Sitting events).
2) Household members use toilets with seats that must be manually adjusted in strictly the up or the down position.
3) The toilet seat is required to be in the up position for all Standing events. (Though errors are assumed to occur).
4) The order of toilet visits is completely random with respect to Standing events and Sitting events.

Four different toilet visit scenarios are therefore possible, with the given respective functions:
a. Standing event occurs after a Sitting event => f(p) = (1-p)*p
b. Sitting after Standing => f(p) = p*(1-p)
c. Sitting after Sitting => f(p) = (1-p)*(1-p)
d. Standing after Standing => f(p) = p*p

Note: The symmetry of "a." and "b.", this is true for all values of p.
Also, the cumulative probability of all four functions totals unity, 1:
= (1-p)*p + p*(1-p) + (1-p)*(1-p) + p*p
= p-p2 + p-p2 + 1-2p+p2 + p2
= 1 + 2p - 2p - 2p2 + 2p2
= 1

E. Evaluating total toilet seat touches by policy:

Sometimes when a toilet is visited we will need to touch the seat. The expected number of touches per visit, T, will vary by policy.
1) “Sitter-Centric”: This policy will require household members visiting the toilet for Standing events (after a Sitting Event and after a Standing event) to touch the toilet seat twice, once to raise the seat, and once to lower the seat. The toilet seat will never need to be touched for Sitting after Sitting events or Sitting after Standing events.
TSC = Two touches for Standing after Sitting events plus two touches for Standing after Standing events
TSC = 2*(1-p)*p + 2*p*p
TSC = p - p2 + p2
TSC = 2p

2) “Shared Responsibility”: This policy will require household members visiting the toilet for Standing events to touch the seat once to raise the seat (and leave it up) for all Standing after Sitting events and household members visiting the toilet for a Sitting event to touch the seat once to lower the seat (and leave it down) for all Sitting after Standing events.  It will never need touched for a Standing after Standing event, or a Sitting after Sitting event.
TSR = One touch for Standing after Sitting events plus one touch for Sitting after Standing events
TSR = (1-p)*p + p*(1-p)
TSR =  p - p2 + p - p2
TSR = 2p - 2p2
We can readily see that 2p > 2p - 2p2 for all values of p > 0.  So the toilet seat is touched more for the "Sitter-Centric" policy when at least one member of the household performs standing events.  The reduction in the expected number of touches per visit is TSC / TSR = 1/(1-p).  Also, TSC - TSR = p2. and TSR = TSC - p2.

F. Error rate:

Some folks might consider touching a toilet seat to be a basic need for doing business, one that washing ones hands negates, and therefore afford no consideration for how often they touch it.  Even so, we can move on to a more negative aspect which is finding pee on a toilet seat.  This part of my analysis required some pause.  At this point I must decide whether it is best to consider the error rate to only Standing events, or to all events. Even more complex is defining exactly what an error is. It is quite simple to recognize an error when you discover one; "son-of-a...".  But calculating specific probabilities of events that occur leading up to and after errors can be complex. All approaches presented me with their own set of difficulties and, without explaining further, I settled on what I feel is an straight forward definition of an error.

I define an error as: Simply failing to ensure that the toilet seat was moved to the up position to perform a Standing event.  This definition asserts that if the toilet seat is left down for a Standing event it will inevitably be soiled 100% of the time be it by direct flow, trickle, sprinkle, splatter, or splash.  In other words the definition gives a zero percent chance that one can successfully perform a Standing event with the seat down and not get even a spot of fluid on the toilet seat.

I also settled on defining the error rate, E, as the probability of an error to all toilet events.  Since errors can only occur during a Standing event, I could express an error rate, e, to Standing events only. To do so would give; e = E/p with e > E.

Note that the concept of errors under the Sitter-Centric policy in which the toilet seat is not put back down after a Standing event was not analyzed.  While this will reduce the actual number of Sitter-Centric touches calculated above, we can ignore it with respect to errors that result in the discovery of residue on the toilet seat since these events neither leave residue nor contributed to the leaving residue.  Additionally, such "errors" are specifically allowed under the Shared Responsibility policy and the purpose of this analysis is to contrast the two policies not see how one might occasionally look more like the other.

G. Defining and evaluating the discovery of an error by policy:

I did not spend too much time discussing the error rate because I believe that it is more useful to calculate the discovery rate of an error rather than the error rate itself since the discovery of an error is the primary concern when visiting a toilet. As I did with the error rate, I express the discovery rate, D, to total toilet visits. Since the discovery of an error can only occur during a Sitting event, I could express a discovery rate, d, to Sitting events only. To do so would give; d = D/(1-p) with d > D.

My definition of the discovery of an error has the following assumptions:

1) The discovery is always associated with the first Sitting event that followed the error.
2) If fluid on a toilet seat dries, it will leave a sticky residue and therefore still be considered discoverable.
3) There can only be at most one discovery per error. That is, the sitter discovering the error is assumed to sufficiently clean the seat (purposefully or latently) such that it will not be possible for subsequent Sitting events to also discover the same error.  Likewise, the sitter will not be able determine or count multiple errors that took place before their Sitting event.  Therefore, successive errors leading up to a Sitting event will result in a single discovery.

First, we will see as we did with toilet seat touches, that the expected number times a household member discovers an error per visit, D, varies by policy. The discovery rate is therefore calculated independently by policy; DSC and DSR.

Calculating the discovery rate proved to be difficult for me after so many years removed from university probability courses. I therefore submitted my calculations to a peer review and believe all calculations are correct.

Note that the discovery rate, D, is always at least a skosh smaller than the actual error rate, E, since there is a probability that more than one error takes place before a Sitting event occurs to discover (and clean) the error(s).

Mathematical Definition of D = The cumulative probability that at least one error occurs in any string of Standing events followed by a single Sitting event.  Since successive errors do not "count" towards discovery (for the purposed of calculating the discovery rate I have already defined that only one discovery results from any number of errors that take place before a single Sitting event) it is acceptable to ignore successive errors and calculate the probability that exactly one error occurred in place of one or more errors.
Given:
Probability(of an Error per all visits) = E.
Probability(Standing event with no Error) = (p-E).
Probability(Sitting event) = (1-p).

1) “Sitter-Centric”: This policy will require household members visiting the toilet for Standing events to put up before they begin, and then put the seat down after they are finished.  Putting the seat down at the conclusion of a Standing event ensures that the seat is always in the down position when a household member visits the toilet for a Sitting event (a definitional requirement of the Sitter-Centric policy).  However, this method also ensures that the seat is down for all toilet visits including Standing events which provides an additional opportunity for an error since the Standing event members are required to raise the seat more often.

DSC = (All combinations of standing visits that result in any number of errors) * (probability that a sitting event occurs after the string standing events)
DSC = (All combinations of standing visits that result in exactly one error) * (probability that a sitting event occurs after the string standing events)  (as we showed above, successive errors can be ignored since they have zero impact on the discovery rate)
DSC = (E + E(p-E) + E(p-E)2 + E(p-E)3 + E(p-E)4 + ......) * (1-p)
DSC= E * (1 + (p-E) + (p-E)2 + (p-E)3 + (p-E)4 + ......) * (1-p)
DSC = E * (1/(1-(p-E)) * (1-p)
DSC = E*(1-p) / (1-(p+E)), or
DSC = E*(1-p) / E+(1-p)

2) “Shared Responsibility”: This policy allows household members visiting the toilet for Standing events to leave the seat in the up position when they are finished. This lessens the chance that an error is made because the seat might already be in the up position when a household member may otherwise have failed to deliberately move the seat to the up position for a Standing event.

As with the Sitter-Centric policy, successive errors cannot increase the discovery rate.
DSR = (All combinations of standing visits that result in exactly one error) * (probability that a sitting event occurs after the string standing events)
DSR = (E + E(p-E) + E(p-E)2 + E(p-E)3 + E(p-E)4 + ......) * (1-p)
Note, because there is a probability that the seat is already up when a Standing event takes place, we clearly have less of a chance of producing errors during a string of standing events.  This results in a probability of discovery (All combinations of standing visits that result in an error) being reduced to only, E, the error rate itself. Put another way, each of the following terms in the expression above; E(p-E) + E(p-E)2 + E(p-E)3 + ....... , are equal to zero.  That is, since the seat is always left up after a Standing event under this policy, there does not exist a possibility of an error occurring during a string of standing events other than the probability of the initial error occurring during the very first Standing event that followed a Sitting event.  Therefore;
DSR = (E + 0 + 0 + 0 + 0 + ......) * (1-p)
DSR = E*(1-p)

We see immediately that DSC DSR / E+(1-p).  So then DSC > DSR  when E+(1-p) < 1. Since E <= p this term can never be > 1.  Let's examine these discovery rates at the limits of p and E.

1) Let p = 1.  DSC DSR = 0 when p = 1.  That makes sense, when there are no Sitting events there cannot be a discovery of an error, however it is usually never the case that p = 1, that is, there should always be Sitting events.

2) Let p = 0.  DSC DSR  = 0 when p = 0.  That makes sense, when there are no Standing event the error rate is zero and there cannot be the discovery of an error.

3) Let p = E.  For any 0 < p < 1 with p = E, then the denominator of the Sitter-Centric discovery rate, E+(1-p) = p + (1 - p) = 1.  So for p = E, DSC DSR = p(1-p).  Again, that makes sense.  If every standing event is an error then for both policies the seat is always in the down position for a standing event and therefore the discovery rates of errors will be identical.  However, it then follows that DSC > DSR for all E < p, which will reasonably be the case for all households.

So, at the limits of p and E there is not difference in the policies.  However, most households do not represent the limits of p and E.  When p < 1 and E <=p then, E+(1-p) < 1 for all p and all E.  (The denominator of the Sitter-Centric discovery rate is less than one which will always make the discovery rate of that policy great than the discovery rate of the Sharred Responsibility policy).

The ratio of the expected decrease in discoveries per visit of DSR to DSC is DSC / DSr = 1 / E+(1-p), where E+(1-p) < 1 for all 0 < p < 1, and all E < p.

H. Summary (Example):

Example: Suppose a household has a mix of members such that Standing events are performed 55% of the time and one-eleventh of those visits result in an error.

So e = 1/11 = E/p.  E = e*p = 55%* 1/11 = 5%.  So the overall error rate, E, to all events is 5%.

Given: p = 0.55 E = 0.05

1) Which Policy has the least number of touches?
Answer: “Shared Responsibility” The "Shared Responsibility" policy will always result in less touches.
Specifically, Probability of touches per visit:
“Sitter-Centric”:                TSC = 2p; Ex. 2(0.55) = 1.100 expected touches per visit.
“Shared Responsibility”:    TSR = 2p – 2p2; Ex. 2(0.55) - 2(0.3025) = 0.495 expected touches per visit.
For the parameters given, the ratio of reduction touches per visit is:  Ratio = TSC / TSR = 1/(1-p); Ex. 1 / (1- 0.55) = 1/0.45  = 2.222. More than twice the reduction in error! (as we also see from 1.100 / 0.495 = 2.222)

2) Which Policy has the least number of error discoveries?
Answer: “Shared Responsibility” The "Shared Responsibility" policy will always result in less discoveries of errors because there is a reasonable probability under this policy that a potential error was thwarted by finding the seat already in the up position.
Specifically, Probability of discovering an error:
“Sitter-Centric”:                DSC = E*(1-p) / E+(1-p); Ex. 0.05*(0.45) / 0.05+(0.45) = 0.0450 discoveries per visit.
“Shared Responsibility”:    DSR = E*(1-p); Ex. 0.05*(0.45) = 0.0225 discoveries per visit.
For the parameters given, the ratio of reduction in the discovery rate of an error is:  Ratio = DSC / DS = 1 / E+(1-p); Ex. 1 / (0.05 + (0.45)  1 / 0.50 = 2.000. Twice the reduction in error discovery! (as we also see from 4.50% / 2.25% = 2.000).

3) Which household members are discovering the errors most often?
Answer: The members that only perform Sitting events. This is obvious, since and error can only be discovered during a sitting event then those who only sit should discover more errors.
Specifically, Who sits in more pee?:
Members who Stand and Sit:   Perhaps these members sit 20% as often as Sitting only members sit. That means they will find pee on the seat 20% as often as well.
Members who only Sit:            These members sit 100% of the time and therefore in comparison to the member above, will find pee on the seat 5 times more often.

4) Finally, is it likely that you found someone else's pee on the seat?
Answer: Yes and No. Members who always sit are incapable of getting pee on the seat and therefore if they find pee on the seat there is a 100% chance it belongs to another family member. These sitting only members should have the greatest desire to select a policy that results in the fewest discoveries of pee on the seat in the first place.  On the other hand, if you are the only member of the household performing Standing events, then you obviously have a zero percent chance of sitting in someone else's pee.

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