Hopes&Fears asked mathematician, probability expert, author, Ph.D Peter Olofsson to spend hours of his time devising a formula for the likelihood of getting pooped on by a pigeon in New York City. Miraculously, he humored us. Read the entire proof of concept or scroll down to the end for the answer!

#### Peter Olofsson, Ph.D

Peter teaches probability and mathematical statistics and is the Chair of the Mathematics Department at Trinity University in San Antonio, TX. He is the author of the books "Probability, Statistics, and Stochastic Processes" and "Probabilities: The Little Numbers That Rule Our Lives."

Hello Peter,

I'm writing from an online publication Hopes&Fears, and I was wondering if you could advise me for a piece. This might be silly: Currently, I am seeking a gamely probability expert to help me devise a formula for how likely it is to get pooped on by a pigeon in New York City.

Should I imagine the number of pigeons as an unmoving cloud over the city, divide by the time they spend in flight and their digestive patterns, as well as factoring in surface area of the average person's shoulders and head? Would this work?

All the best,

Whitney Kimball
City editor
Hopes&Fears

Hi Whitney,

Ah, but it's not silly at all! According to Seinfeld, we have a deal with the pigeons: They get out of our way when we drive and we look the other way when they poop on statues. If they poop on us, I think the deal is off. Anyway, as with any problem in mathematical modeling, it can be done in many different ways, and however one does it, there has to be a lot of simplification.

One easy way is to consider a person with head and shoulder area a, moving about in a region of area A. As one pigeon takes one poop-shot at the person, the probability of hitting is a/A.

Naturally, this will be a very small number, but if we have many pigeons taking many attempts, the probability of a hit increases. Here's how we can compute the probability that there is at least one poop hit: The probability of a single hit is a/A. This means that the probability of a single miss is 1-a/A. [Note: Mathematically, probabilities are expressed as numbers between 0 and 1. In daily life, we often use percentages, so "a probability of 0.25" is the same as "a 25% chance," and so on.]

Now, if 2 pigeons try to hit you (or if one pigeon tries twice), the so-called multiplication rule for independent events tells us that the probability that they both miss is the product of the individual probabilities to miss. For example, if each has a 10% chance of hitting you, then each has a 90% chance of missing. The chance of both missing is 90% of 90% which is 81%, so the probability they both miss is 81%. That means that the probability that at least one of them hits you is 19%. Note that "both miss" is the opposite of "at least one hits" so the two probabilities must add up to 100%. Mathematically, the probability of at least one hit from two attempts is:

1-(1-a/A)*(1-a/A)

which is bigger than the probability a/A of hitting in one attempt.

Now let us denote by N the number of pigeon attempts to hit you. Thus, this is the number of pigeons multiplied by the number of times each takes a poop attempt at an innocent pedestrian. By the same rule as above, the probability of at least one hit is

1-(1-a/A)*(1-a/A)*...*(1-a/A)

where the factor 1-a/A is multiplied by itself N times. In more compact mathematical notation:

1-(1-a/A)^N

[one minus (1-a/A) raised to the power N]. It is a well-known mathematical fact that this can be approximated by

1-exp(-N*a/A)

where "exp" denotes the exponential function, [exp(x)=e raised to the power x].

Obviously, there are a lot of facts and circumstances that need to be taken into account. For example, I don't know how prone pigeons are to pooping in flight, but judging from statues and rooftops, they tend to do it while sitting.

Cheers!
Peter

Hi Peter,

Yes, there are a lot of factors to consider... I've just put together a few figures and was wondering how to fit it into the equation:

Roughly 1 million pigeons live an area of 469 mi² (1,214 km²) area or 2,476,320 square feet.

Pigeons spend an estimated 21% of the day perching or flying.

Therefore, 210,000 pigeons are in flight at any given time.

This means that in a 10 square foot space, there are an average of 1.17 pigeons at any given time of the day.

Pigeons spend about 20% of their day pooping.

Therefore, in every given 10 square feet of New York, you have .234 pooping pigeons flying around.

Surface area of average human head and shoulders: 145.43 square in = 1 square foot. (I based this on the sketchandcalc below, which is scaled to the average human head circumference of 23in).

So, if a= 1 (the head and shoulder area) and A= 10 square feet (the flight area), how might I factor in the rate of .234 pooping pigeons in that area?

All the best,
Whitney

Surface area of average human head and shoulders: 145.43 square in = 1 square foot

how many pigeons are in NYC?

Nobody knows how many pigeons are in NYC; not the National Wildlife Research Center, the pigeon birth control company Ovo Control, the Audubon Society (which conducts an annual bird count), or the Brooklyn Bird Club. Pigeons can breed up to six times a year, depending on food supplies. Most articles have simply put the estimate between 1 and 7 million birds.

Whitney,

If we use A=1,000 sq ft instead, we get 23 pigeons which is easier to grasp than a pooping quarter-pigeon.

There's one problem with the data you present. What we really need to know is how often a pigeon poops.

I don't quite understand what it means that a pigeon spends 20% of its time pooping. We need to think of a pooping event as a dart throw, instantaneous in time. If I say I spend 20% of my time throwing darts, we don't know what that means in terms of how often I throw. Hope this makes sense. If you can get that information, I think we're ready to go.

Also, I assume they only poop during the 21% of time they perch or fly, and the rest of the time they eat or sleep. Thus, at any given time, there are on average 210,000 pigeons who are ready to poop. Obviously the actual number varies over time, some due to randomness, but also due to diurnal regularity (coming home to roost in the evening, sleeping at night, etc).

Best,
Peter

Peter,

Ah, this makes sense. The pooping number comes from a show bird owner's estimate that pigeons poop every 10-15 minutes, but I just made a few dozen calls to vets and bird stores to confirm. Reports range from: 10 times a day; between 15 and 50 times a day; every 3 minutes; at least every 40 minutes; at least every 4 hours; every 15 minutes; “all day long"; every 3 hours; every time it eats; "there's no way to time it because it depends on stress, how often it eats, and its diet".

I just set the hypothetical poop rate at every 12 minutes for the sake of an even number, and reports of compulsive over feeders in NYC.

And yes, the 21% of potential poop time comes from a study which finds that pigeons spend 11% of the day chasing other pigeons, 7% perching, and 3% flying (the rest is sleeping, eating, walking, and other on-the-ground or motionless activities).

Best,
Whitney

Whitney,

Very good, now we're in business. Let us assume that pigeons poop on average 5 times per hour. In the original formula

1-(1-a/A)^N

the number N is now the number of pigeons multiplied by the number of hours you are exposed, multiplied by 5. So if we use a=1 and A=1,000, we have N=23*5*t where t is the number of hours. For example, if you are out for two hours, we get N=23*5*2=230 and the probability of being hit at least once is

1-(1-1/1000)^230 = 0.20

a 20% chance. The breaking point where it is more likely than not to get hit is a little over 12 hours (trial and error with different values of N; can also be computed by using so-called "logarithms").

A few notes from the professor in me:

There are essentially two different ways of computing a probability like the one you are asking for. The first is the way we have done it, using a mathematical model where one makes certain assumptions (pigeons fly around randomly, are evenly distributed over an area, etc). The model has some parameters like a, A, and N, and once these can be estimated from data, we compute our probability. The other way is to estimate the probability directly by using data on the event we are interested in. Here, we could simply ask a number of people if they have been pooped upon (akin to an opinion poll), or we could do an experiment and send out a number of people to walk around for some time and then inspect them for poop. This could also be a way of validating our model, to make sure observed numbers are reasonably consistent with our predictions. For example, if 100 people are out for 2 hours, our model says that we expect 20 of them to be hit by poop. If we send out 100 people and the number of unlucky bastards deviate very much from 20, such as 0 or 50, we would suspect our model is not that great.

In reality, N must be an integer. In our formula, however, it does not necessarily come out to one. For example, if you are out for half an hour,

N=23*5*0.5=57.5

This is because our N is really an average, or so-called "expected value" whereas the real N is a "random variable," meaning that its value varies. A pigeon poops on average 5 times per hour but in a given hour, the actual number of poops may be 0,1,2... with different probabilities. More sophisticated modeling can be done using so-called "Poisson processes" to describe pooping events.

Enough lecturing!

Will you also provide tips on how to get rid of pigeon poop stains from your favorite linen jacket?

Best,
Peter

Peter,

So according to this formula, if pigeons poop at the very low estimate of once every four hours…

N=23*.25*2 = 11.5

1-(1-1/1000)^11.5 = .0114

Therefore the chance of getting hit in 2 hours would be

.0114, or 1.14%?

Or, if you spent one minute during your daily commute crossing under a dense pigeon congregation of 100 pigeons per 100 square feet (under, say, the 46th Street, 52nd Street and 61st Street subway stations) the equation would be:

N=100*5*.0166 = 8

1-(1-1/100)^8=.07, or 4%

So in that case, frequent walkers have a very good chance of encountering a pooping event about once a month? (This would be consistent with the 2013 War on Pigeons rhetoric).

As for the stain… I made a few phone calls and am finding a curious disparity. Linen companies all advise machine washing on a cold gentle cycle and air dry. Dry cleaners say that the only possible way is to dry clean it. Weird, huh? I also got an answer from Becky, the creator of the cleaning megablog Clean Mama...

"I would suggest that you let it dry and scrape it off, then launder it or dry clean according to directions :)"

I personally prefer the baking soda and tonic method.

Whitney

#### Increased changes of getting pooped on by wearing a hat

↑ Head and shoulders surface area for a head with a circumference of 23 in= 145.43 SQUARE IN = 1 SQUARE FOOT

↑ Surface area increased to 153.31 square INches = 1.06 SQUARE Feet, which makes the new equation:

1-(1-1.06/1000)^230 = .21

A hat increases likelihood of poopage by 1%.

Whitney

The N will always be #pigeons x time x poop rate.

Best,
Peter

## 20%

You are 20% likely to be pooped on by a pigeon within 2 hours, assuming you are walking nonstop through New York City, which contains a population
of 1 million pigeons who poop at a rate of every
12 minutes

Editor's Note: Our calculations were revised in light of miscalculation of the surface area of human head and shoulders.