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