On Birthdays and Monkey Poo

It’s my sister Laurie’s birthday today, and much as it pains me to say it, it’s never easy being the four-years-younger brother of a super genius.  When you’re graduating middle school, she’s graduating high school.  When you’re graduating high school, she’s graduating college.  When you’re getting a BS from MIT, she’s getting a PhD from Harvard.  When you publish your first book, she’s giving a TED talk and getting tenure at Yale.  Even today as she grows ever older and more decrepit, turning an age I can remember our mom and dad being, I can still feel that familiar tingle of our (usually) healthy sibling rivalry.  If there's one thing I take solace in, it's that I've never had to worry about getting monkey poo in my hair.  What’s the probability that at some point in her career as a primate researcher, Dr. Santos has gotten monkey poo in her hair?

Cayo Santiago is one of the places my sister does research on primates.  Located southeast of Punta Santiago, Puerto Rico, the 600 m by 400 m (240,000 m²) island is home to about 950 Rhesus monkeys.  As can be seen from the photo on the right, Cayo is a fairly lush island featuring lots of trees monkeys can climb.¹  I’ll assume a typical monkey spends about 50% of its time climbing, meaning that at any moment there are about 475 monkeys in the trees.

Each year, Laurie generally spends about 2-4 weeks on Cayo, and she has done so for about 10 years.  If she works 8 hours per day, she’ll have spent up to 93 days on the island.  Assuming a Rhesus monkey’s digestive schedule is similar to that of a human, each monkey probably “uses the bathroom” at least once per day.  From this and the information above, we can estimate the total number of high altitude monkey poos taken while Laurie has been on the island

(# of poos per day per monkey) · (# of monkeys) · (# of days)

= (1.0 poo per day per monkey) · (475 monkeys) · (93 days)

= 44,175 poos.

From a bird’s—er—monkey’s-eye view, her hair will appear to have about 1.0 ft² of area.  Our monkey bombardier will have to hit this spot if Laurie is to spend the rest of her night shampooing.  From this we can calculate the probability of a random poo landing in her hair,

P = (area of hair) / (total area)

= (1.0 ft²) / (240,000 m²)

= 3.9×10^-7.

The probability of a poo not landing in her hair is, of course, 1 – P.  The probability that none of the 44,175 monkey poos have landed in her hair is (1 – P) ^44,175 ~ 0.98.  This means there’s a 2% chance that a monkey has poo-ed on my sister’s hair at least once.

Happy Birthday, Laurie!  You’re as good a science role model and an even better sister than I could reasonably have hoped for. 

[1] For simplicity, I’ll assume that 100% of the island is covered in trees.