# Umbrellas to work, or why engineers should get married

When I was working in Yahoo at Suntec City, I used to take the MRT to work every day and my route is completely sheltered right to the doorsteps of the office so the weather didn’t really have much effect on me. I still take the MRT to the office now that I am at Garena, but there is a short distance from the MRT station to the office, about 5 minutes of walking, where it is subject to the elements.

Now I don’t take any umbrellas along with me, so with the recent change in weather (it has been raining cats and dogs and all sorts of mammals in between) I have been caught a few times in some unpleasant wetness. It is during one of these wet commutes from home to office I was thinking of ways of making sure that will not happen again.

The idea was like this. I will place some umbrellas at the office and also at home. Whenever I leave the house or the office and if it is raining, I will take the umbrella to the office or back home respectively. Naturally if it’s not raining then I wouldn’t want to look like a dork and carry one.

Therein lies the problem. Let’s say I keep 1 umbrella at the office and another at home. If it rains when I’m leaving the office, I will carry that one home, leaving 2 at home and none at the office. If it rains the next day when I leave home it’s ok, I will just take an umbrella to the office, making it 1 – 1 again. If it doesn’t rain and therefore I don’t take any umbrellas to the office, I will be left with 2 – 0. If it happens to rain when I’m leaving the office I’m stuck with getting wet again.

What if I have 2 umbrellas at the office and 2 at home? In this case I will be caught wet only if the same occurrence (rain when leaving the office followed by no rain when leaving the house) happens 2 days in a row, which probability is lower than if it happens just once. Still, that can happen and I want to get assurance that the risk of getting wet is really minimal. In that case, the question becomes, how many umbrellas should I leave at the office and similarly at home such that the probability of getting wet is negligible? Note that it doesn’t really matter if one place has more umbrellas than the other since it will be the same as the lower number of umbrellas in the long run.

To find out, I used a popular (and really pragmatic and therefore very ‘engineering’) method of finding probabilities — the Monte Carlo simulation method. The name sounds fancy but it’s really just an algorithm that uses repeated random samples to find the answer. In our case, what we want to do is to find out, over large sample, the average number of trips I will make before I run the risk of getting wet.

To find this, I will need to inspect the probability that it will rain at any given day. For easy of calculations, I use a probability between 1% chance that it will rain and 99% chance that it will rain. Why don’t I include 0% or 100%? This is because if it is 0% it means it will never rain and so I will never need to use an umbrella, and if it 100% it will always rain and therefore I will always bring an umbrella back or forth.

On top of that, I will run the simulation for the case where I have 2 umbrellas at home and at the office, up to 50 umbrellas at home and at the office. To give a large enough representation, I run this over 1000 iterations (more will be better but it will take too long to run). This is the Ruby code for the simulation.

```require 'rubygems'
require 'faster_csv'

umbrellas_range = 2..50
probability_range = 1..99
@location = :home

total_trips = {}
iterations = 1000

def walks_to(loc)
location = loc
@num_of_trips += 1
end

def office?()  @location == :office; end
def home?() @location == :home; end
def raining?(probability) rand(99) < probability; end

FasterCSV.open('umbrellas.csv', 'w') do |csv|
csv << [''] + probability_range.to_a
umbrellas_range.each do |umbrellas|
probability_range.each do |probability|
total_trips[probability] = 0
iterations.times do
@num_of_trips = 0
wet = false
home = umbrellas
office = umbrellas

while not wet
if home?
walks_to :office
if raining?(probability)
if home > 0
home -= 1
office += 1
else
wet = true
end
end

elsif office?
walks_to :home
if raining?(probability)
if office > 0
office -= 1
home += 1
else
wet = true
end
end
end
end

total_trips[probability] += @num_of_trips
end
end

row = [umbrellas]
total_trips.sort.each { |pair| row << pair[1]/iterations }
csv << row
end
end
```

The Ruby program writes to a simple CSV file called umbrellas.csv with the first row being the probability, and the first column the number of umbrellas at each location. If you chart each row with the y-axis being the number of trips and the x-axis being the probability of rain from 1% to 99%, you will find a chart like this:

Now that I have the average number of trips that I will make before I get wet, I want to actually see what that means for Singapore. To do this, I went to the National Environment Agency’s website and dug out the weather statistics for Singapore.  There is a statistic that gives the average number of rainy days in any given month for the past 118 years. I take that for each month and divide that with the number of days in that month to derive the monthly probability of rain as in the following table:

 Month Probability of rain January 48% February 39% March 45% April 50% May 45% June 43% July 42% August 45% September 47% October 52% November 63% December 61%

Finally I match that with the Monte Carlo simulation I made earlier. The average number of trips for each month, must be between 56 to 62 days (each trip is one way only, so every day is 2 trips). For example, the statistics shows that the wettest months are in November and December which is 63% and 61% probability of rain respectively. This means I will need to have 38 umbrellas at each location for each of these 2 months (please check your umbrellas.csv file; open it up with Excel). In February where the probability of rain is the lowest (39%), I will only need 23 umbrellas at each location to almost guarantee that I will not get wet (the whole deal assumes at the end of each month, I replenish each location with the necessary number of umbrellas, of course).

That evening when I proudly told my wife my findings at home, she stared at me for while (frostily if I might add) then gave me a small folding umbrella, which I now carry in my laptop bag.

And that is the reason why engineers should get married.

(I have been inspired by this book — Digital Dice : Computational Solutions to Practical Probability Problems, by Paul J. Nahin)