One of my initial goals for 2011 was to get my feet wet with Python, but after the last (and excellent) San Francisco F# user group meetup, dedicated to F# for Python developers, I got all excited about F# again, and dug back my copy of Programming F#.

The book contains a Sequence example which I found inspiring:

open System

let RandomSequence =
  let random = new Random()
  seq { 
    while true do
    yield random.NextDouble() }

What’s nice about this is that it is a lazy sequence; each element of the Sequence will be pulled in memory “on demand”, which makes it possible to work with Sequences of arbitrary length without running into memory limitation issues.

This formulation looks a lot like a simulation, so I thought I would explore that direction. What about modeling the weather, in a fictional country where 60% of the days are Sunny, and the others Rainy?

Keeping our weather model super-simple, we could do something along these lines: we define a Weather type, which can be either Sunny or Rainy, and a function WeatherToday, which given a probability, returns the adequate Weather.

type Weather = Sunny | Rainy

let WeatherToday probability =
  if probability < 0.6 then Sunny
  else Rainy

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Today is Friday the 13th, the day when more accidents happen because Paraskevidekatriaphobics are concerned about accidents. Or is it the day when less accidents take place, because people stay home to avoid accidents? Not altogether clear, it seems.

Whether safe or dangerous, how often do these Friday the 13th take place, exactly? Are there years without it, or with more than one? That’s a question which should have a clearer answer. Let’s try to figure out the probability to observe N such days in a year picked at random.

First, note that if you knew what the first day of that year was, you could easily verify if the 13th day for each month was indeed a Friday. Would that be sufficient? Not quite – you would also need to know whether the year was a leap year, these years which happen every 4 years and have an extra day, February the 29th.

Imagine that this year started a Monday. What would next year start with? If we are in a regular year, 365 days = 52 x 7 + 1; in other words, 52 weeks will elapse, the last day of the year will also be a Monday, and next year will start a Tuesday. If this is a leap year, next year will start on a Wednesday.

Why do I care? Because now we can show that every 28 years, the same cycle of Friday the 13th will take place again. Every four consecutive years, the start day shifts by 5 positions (3 “regular” years and one leap year), and because 5 and 7 have no common denominator, after 7 4-year periods, we will be back to starting an identical 28-years cycle, where each day of the week will appear 4 times as first day of the year.

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A new method of forecasting has been brought to my attention: Paul. Paul is an English-born octopus, currently living in Germany, and has been predicting with high accuracy the results of the German soccer team, by picking between two boxes marked with the flag of the two competing teams:

How unlikely is it that Paul would have such a string of correct forecasts? Pretty unlikely. If you assume that Paul’s picks were completely random, with a 50% chance of correctly calling the winner, the probability of making 11 good calls out of 12 comes down to 0.29%. Does this mean Paul is the next big thing in forecasting? It’s possible, but I don’t think so (this said with all due respect to Paul and his skills). Leonard Mlodinow, in his excellent book, The Drunkard's Walk, makes the following comment:

This example illustrates an important point: even with data significant at, say, the 3 percent level, if you test 100 nonpsychic people for psychic abilities […], you ought to expect a few people to show up as psychic.

In other words, if a phenomenon is random, you should typically see the “average” case regularly, but you should also see highly unlikely cases happen from time to time – observing such a rare occurrence doesn’t contradict the randomness of the phenomenon. Or, in the words of the French poet Mallarmé, Un Coup de Dés Jamais N'Abolira Le Hasard (A throw of the Dice will Never Abolish Chance).