Estimation: Portable Restrooms

The basic idea here is: Let's say that a 'typical person' spends 20% of their time in some activity (e.g. eating, studying birds, or perhaps scratching mosquito bites). Then for a larger number of people, roughly 20% of the people in that group are engaged in that activity at any one time:

So if the world's population is roughly 6 billion, I would estimate that at any moment, roughly 1/3 of them--2 billion of them--are asleep.

Your experience

Use yourself as 'typical' to figure out what fraction of *your* time *you* spend in the bathroom.

For bathrooms, this fraction is the ratio of the time you spend in the bathroom to the total time you're awake.

Use your own experience as 'typical'. Guess, or estimate how much time you spend in the bathroom everyday (excluding showers and working on your appearance). Is it closer too...

Length of a day

Of course, we probably don't think of large portions of a day in minutes, but rather in hours. So estimate in hours, and then convert to minutes. Multiply hours by the number of minutes in one hour.

Fraction of your day in the bathroom

Now you should have the numbers, all converted to minutes, to be able to calculate this number for yourself:

Bathroom fraction = $\frac{\rm bathroom\ minutes\ in\ a\ typical\ day}{\rm total\ waking\ minutes\ in\ a\ day}.$

Now, to estimate the number of portable restrooms you'll need, take the total number of people, and multiply by your "bathroom fraction". This is the number of people interested in using the bathroom at the same time.

If you stick with this number as your estimate, that would mean that all the "portable restrooms" would be continuously occupied all day long. To avoid long waits, perhaps you should add a few more.

Write up the assumptions that you make, and finally, how many restrooms you settled on renting for this situation:

You are organizing a political rally and expect 100,000 people will come. How many porta-potties should you arrange for?

Example:

I'll estimate 10 minutes in the bathroom during my waking hours.

I'll estimate that I'm awake 16 hours during a typical day:

`16 hours times frac(60 minutes)(1 hour) = 960 minutes approx 1000 minutes`

So, the fraction of time I estimate that I'm in a bathroom during a typical day is:

`10 / 10000 = 1/100 = 1%`

Now, we'll make the intellectual leap that at any one moment in time, the fraction of a large number of people engaging in some activity is the same as the fraction of time that one person spends on that activity.

So, from my personal time estimates, I'm estimating that 1/100 of the number of people in a large group would typically be in a bathroom at any point in time. In particular, 1/100 of 100,000 is

`100000 \times \frac{1}{100} = 1000`

So, I need 1000 port-a-potties.

Your estimates for the number of toilets for 100,000 people:

estimates

Virginia Children's Festival: L. Weinstein says there are about 40-50 port-a-potties for this annual Norfolk event which attracts 3000 people. 50 would be `50/3000 = 0.0167 approx 1.7%` or 1.7 toilets for each 100 people.

New York City marathon: 1660 toilets in the starting area for 29,000 people: 5.7 toilets for each 100 people.

Lawrence Weinstein counted somewhere around 50 toilets for the 3000 people that come for the Virginia Children's Festival-- a ratio of 1 toilet per 60 people.