Man, Woman, Toilet Seat

Whether or not you believe there’s a Battle of the Sexes, there’s one bit of eternally contested territory: what to do with the toilet seat. Casual observation suggests that the dominant view at the moment is that men should always  put it down or leave it down (PIDOLID) when they finish, because women always need it to be down. There’s a superficial ring of good sense to this maxim, but detailed calculations can turn up surprising conclusion. I thereby propose to bring science to bear on the question. I will model the use of a public toilet of the unisex, one-occupant variety, in a locale such as your neighborhood Starbucks. This will call for some assumptions, but I will argue that the results are reasonably robust under changes of those assumptions.

First, the obvious: if everyone followed the PIDOLID rule, women would never need to change the position of the toilet seat. Men, however, would—every time they did #1. This casts a suspicious light on the maxim. However, in the spirit of careful inquiry, let us dig deeper.

Suppose everyone followed the path of least resistance and left the seat where it was when they finished. What would things be like then?*

I will suppose that men and women visit the Starbucks toilet independently and with equal frequency. That means the probability that the next visitor will be male is .5, as is the probability that the visitor will be female. I will also assume that whether the visitor executes a #1 or a #2 is independent of the sex of the visitor. In other words, if all you know is the sex of the person entering the restroom, this provides no information about the function s/he is about to execute. Finally, we need information about the frequency of Nature’s two kinds of calls. I will assume that 4 out of 5 are for #1, so that the probability of a #1 performance is .8, with the probability of a #2 performance therefore .2

Now we need these two probabilities:

  1. The probability that the visitor is a male who will have to adjust the seat position
  2. The probability that the visitor is a female who will have to adjust the seat position

Let us begin with 1. The event in question occurs in three circumstances:

a) The visitor is male AND performs #1 AND the previous visitor was female

b) The visitor is male AND the performs #1 AND the previous visitor was a male who performed #2.

c) the visitor is male AND the visitor performs #2 AND the previous visitor was a male who performed #1.

Because of our independence assumptions, these probabilities are:

a) .5 x .8 x .5 = .2

b) .5 x .8 x (.5 x .2) = .04

c) .5 x .2 x (.5 x .8) = 04

Adding, we get .28. That is: 28% of the time, the visitor will be a male who need to adjust the seat position. The conditional probability now follows quickly. Under this regime, the probability  that you will need to adjust the seat given that you are a male is .28/.5, which is .56, or 56%.

Now let us perform the parallel calculation for female visitors. Assuming everyone leaves the seat where it was at the end of business, and assuming that the assumptions we have made are correct, there is only one circumstance in which a female visitor will need to adjust the seat position:

d) The visitor is female AND the previous visitor was a male who performed #1.

Once again, using our independence assumptions, this is an easy calculation. We have

d) .5 x (.5 x .8) = .2

That is, 20%. The conditional probability that the visitor will need to adjust the seat given that the visitor is female is .2/.5 = .4

Thus: If everyone leaves the seat where it was when they finished, men will need to adjust the seat 56% of the time, women 40%.

The assumptions are not based on actual measurement; my local Starbucks would frown on standing around outside the toilet for prolonged periods, let alone asking patrons what task they just performed. The assumption that type of task is independent of the sex of the performer is perhaps open to some doubt, but it seems unlikely that it’s too far off the mark. In any case, we can check the robustness of the results under adjustment of the frequency assumptions.

Suppose, for example, that the ratio of #1 to #2 is 3:1 rather than 4:1, giving us p(#1) = .75. Then if men and women visit the convenience equally often, men will need to adjust the seat position 56.25% of the time, to women’s 37.5%.

Suppose that we keep the #1 to #2 ratio at 3:1 and assume that men make 60% of washroom visits. Men will still need to adjust the seat position 52.5 of the time, versus 45% for women. (Seat of the pants guessing suggests that the 3:1 ratio is too low; perhaps a grant application is in order.) In any case, it is perhaps worth noting that if 2/3 of bathroom breaks are taken by men, and if the #1 to #2 ratio is 3:1, then men and women will each need to adjust the seat position 50% of the time. (The policy implications of this fact are not entirely clear.)

What we see, then, is that under reasonably robust assumptions, if everyone leaves the seat where it was when they finished, men will need to move it more often than women. Whether this is a welcome result will no doubt vary with the reader. However, we should all be able to agree on two things: men who leave the seat down when performing #1 are to be roundly condemned, as are women who hover and spray.

*A common complaint against members of my sex is that this is just what they actually do.

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