Economists have been getting a bad rap lately. We get blamed for everything going wrong in the world. From Wall Street corruption to unemployment to climate change, it’s somehow our fault one way or another. We are cast as the cruel and greedy, heartless plotters who spend all day machinating global dominance. The Considerate Economist is here to show you just how considerate economists can be by using our tools towards good and harmony.

The perennial battle of the sexes occurs over a number of household issues, but few arguments get as heated as they do about leaving the toilet seat down. One day after getting an earful for leaving the seat up (vertical position) I went to the gym. When I arrived I walked into the unisex bathroom and the seat was up. I noticed that nearly everyone in the weight room and basketball court was male, which possibly explained this shocking phenomenon (Don't these guys know better?). But if there were more women at the gym at the time, would the seat have been up?

I wondered how many women relative to men need to be present that the toilet seat should be put in its horizontal (or “down”) position. This ratio, I decided, would determine whether one should put the seat up or down, and would absolve me from being considered rude under certain circumstances. You might be wondering what men/women ratio is that would justify deliberately leaving the plastic seat pointing at the heavens even when women are present. Read on.

Using a basic threshold calculation methodology, I will answer the gripping question of when does one put the toilet seat down, and when does one leave it up.

Here’s the mathematic setup:

Let's assume that women need to put the seat down 100% of the time to go to the bathroom. Mathematically, the conditional probability is P(D|W)*=1.

P(D|W)=Probability of toilet seat down given a woman walks into the bathroom to use the toilet = 1.00. B simply means "down".

Men, on the other hand probably put the toilet seat down once for every 5 visits to the bathroom. Therefore, the conditional probability of putting the seat down given the person is male is P(D|M)= 0.20 (this means that we need the seat down with a 20% likelihood. This figure is an estimate and may require empirical verification).

Let's say that out of 10 people present in a house with a unisex bathroom, how many women must be present so that the likelihood that the toilet seat must be placed down exceeds 50%?

P(M) is simply the probability of a male walking into the bathroom. P(W) is the probability of a female walking into the bathroom. We use variables for simplicity:

P(W)=µ is the probability of a woman walking into the bathroom

P(M)=π s the probability of a man walking into the bathroom

Mathematically: P(D|W)P(W)+P(D|M)P(M)>.50

This is an expected value of putting the seat down.

In solving this problem, we seek the threshold beyond which the probability of needing to put the seat down exceeds 50%.

µ+π=1 This simply means that the probability of either a woman or man walking into the bathroom is 1 (Granted, gender is not always a binary variable. The exact same setup would work if gender ambiguous people were in the problem and would not change the mathematics at all.).

µ+π=1⇔µ=1-π ⇔π=1-µ

P(D|W)P(W)+P(D|M)P(M)>.50 where P(D|W)=probability of needing to put the toilet seat up given a women walks into the restroom; P(A|M) is for men.

P(D|W) = 1, P(D|M) = 0.20, P(W) = µ, P(M) =π

µ+0.20π>0.50

By substitution: µ+0.20π => µ+0.20(1-µ) => µ+0.20-0.20µ => 0.80µ+0.20

0.80µ+0.20>0.50

0.80µ +0.20 -0.20 > 0.50 - 0.20 = 0.30

0.80µ>0.30

µ>0.30/0.80

µ>3/8.

Therefore, the proportion of women to men must exceed 3/8 (which equals 0.375) to justify putting the toilet seat down in a unisex bathroom. In other words, when there are more than 37.5% women present, the likelihood that the next person walking into the bathroom will need the seat down is greater than 50%. Therefore, the considerate thing to do is put the seat in the "down" position**.

I hope this brief foray into bathroom economics has been both insightful and useful.

- The Considerate Economist

*The “P” means “Probability”, and the stuff inside the parenthese means “something given something”. The vertical bar means “given”. An easy example is for coin tosses: P(H|T)=P(T|T)=0.50 means “probability of a heads or tails landing up when tossing a coin is 50%.

** Of course there are scenarios that will change the probabilities of proper seat position, so make sure to use your judgement. For example, if the only woman present is standing behind you in line for the bathroom the decision rule ought be be obvious. This analysis works when there is no information regarding who will use the bathroom next.