The Warm Equation

(Prompted by seeing a bird take a huge dump while flying across my lawn.)

Prior work:

• http://blog.minitab.com/blog/fun-with-statistics/poisson-processes-and-probability-of-poop
• http://www.hopesandfears.com/hopes/now/math/215059-pigeon-poop-probability-nyc
• http://dilbertblog.typepad.com/the_dilbert_blog/2007/09/odds-of-being-s.html
• https://whatif.xkcd.com/11/
• https://www.reddit.com/r/theydidthemath/comments/32y3ov/request_what_is_the_chance_you_will_get_shit_on/

Here's my shot:

1. the continental USA is 8080474 km² or 8080474000000 m²
2. there are ~10b or 10000000000 birds in the USA or 0.001 per square meter (10000000000 / 8080474000000)

3. the average person spends ~6% of each day outside:

   1. "The National Human Activity Pattern Survey (NHAPS): a resource for assessing exposure to environmental pollutants"

      NHAPS respondents reported spending an average of 87% of their time in enclosed buildings and about 6% of their time in enclosed vehicles.

      100%-(87%+6%) = 7% outside each day

   2. "An overview analysis of the time people spend outdoors"

      The median times per day outdoors during weekdays and weekends gave pooled estimates of 1·04 and 1·64 h

      So ((1.04/24)* 5 + (1.64/24)*2) / 7 = 0.05 or 5% of each day

   Take the average, 6%

4. birds poop all the time:

   1. http://www.birdchannel.com/bird-diet-and-health/bird-care/bird-poop.aspx

      "The number of times a bird poops in a day loosely correlates to its size,” Zielezienski said. "Smaller birds eliminate more frequently than larger birds. A budgie may excrete 40 to 50 times in a day, whereas a macaw may only go 15 or 20 times.” Burkett said that birds poop so frequently because they have a high metabolic rate and process food quickly. "In order to fly efficiently and quickly, birds need to carry as little weight as possible, so they evacuate their cloaca frequently.”

   2. http://www.nytimes.com/2008/07/08/science/08qna.html

      ...it has been estimated that depending on diet and season, a goose defecates from around once an hour to once every few minutes, for a total of a pound or more a day of droppings from a large goose.

   Seems safe to say that the average bird will poop something like 30 times a day.

Assuming independence, average probability etc,

• each person has 0.06*24*60*60=5184 seconds daily exposure
• birds poop every 2880 seconds on average, so probability of 1/2880 each second or a poop probability each second of 0.001 * (1/2880)
• thus, probability of surviving 1 day without poop is (1 - (0.001 * (1/2880))) ^ 5184 = 0.9982016187
• or in lifetime chance over 78.74 years (average American life expectancy 2012): (1 - (0.001 * (1/2880))) ^ (78.74 * 365 * 5184) = 3.411896044e-23 ~= 0, which means a lifetime risk of ~1.

(This is probably too high because many birds poop upon takeoff and presumably are systematically far from humans at that point, but it's also counterbalanced by things like cities having inordinately large pigeon/seagull populations and pooping upon people repeatedly because human density in cities is so high. And hey, here on /r/estimation we're happy if we're within an order of magnitude, and given how many people report being pooped on at some point online...)

Ultimately, we are doomed to be attacked by the birds.