# the coconut puzzle

There is a saying in Malayalam, "thenga chadikilla" that translates as "The coconut will not betray you". The typical coconut tree stands anywhere between 50 to 80 feet tall and the unhusked nuts are at the top weighing a few pounds each. As they hang, it is conceivable that some may fall posing a hazard to people on the ground. Whoever founded this idiom was probably amazed at the fact that there are so many coconut trees in Kerala, yet so few accidents related to falling coconuts.

I thought it would be a good idea to figure out what exactly is the probability of this accident under reasonable, simplifying assumptions :).

Assumptions:

- All coconuts randomly fall. This isn't 100% true of course. Nuts are ready for plucking every 45 days.
- People walk randomly in these cultivatable areas. Again, not always true. People are generally conscious of where they are.

Some facts:

- Kerala makes approximately 6 billion coconuts every year.
- Total arable land where coconut trees exist is approximately 9 x 10^11 sq meters.
- There are 3.5 million families earning a livelihood from coconuts, assuming 4 individuals per family, there are 14 million people at "risk".

We can assume that coconuts fall following a Poisson distribution over this cultivatable space. The Poisson distribution looks as follows,

where lambda is the average number of coconuts falling per square meter per day and k is the number of times the event occurs, in our case k=1.

Plugging in the values we can arrive at an average estimate for lambda as shown below.

Using a similar technique, we can find the average number of people per square meter of cultivatable land as follows,

The probability that someone gets hit by a falling coconut is the product of the probability of a person being in an area the same time a coconut falls in that same area (shown below)

Ignoring the exponential terms as they are almost 1, the probability of a coconut related accident works out to be

Clearly this is small :).

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