[mag487]

It may be a useful sanity check to simulate the experiment multiple times to see the result:

  def choose_coin
    rand < 0.001 ? 1 : 0.5
  end
   
  def flip_coin(coin)
    rand < coin ? :heads : :tails
  end
   
  def all_10_heads?(coin)
    10.times { return false if flip_coin(coin) == :tails }
    true
  end
   
  next_flip = { :heads => 0, :tails => 0 }
   
  1000000.times do
    coin = choose_coin
    next unless all_10_heads? coin
    next_flip[flip_coin(coin)] += 1
  end
   
  puts next_flip[:heads].to_f / (next_flip[:heads] + next_flip[:tails])

[jules]

What's really interesting is that there are probabilistic programming languages where you can write a program that does a simulation just like you did, but the execution engine can compute the probabilities exactly and much faster too. It does this by computing along all possible paths in the program, and keeping track of the probability mass of each path, and then summing them all up in the end.

http://en.wikipedia.org/wiki/Probabilistic_programming_langu...

[crntaylor]

Completely shameless plug for a tiny probabilistic programming language that I wrote as an embedded DSL in Haskell:

https://github.com/chris-taylor/hs-probability

The code that solves this problem is:

  solve = do
    coin   <- choose (999/1000) fair biased
    tosses <- replicateM 10 coin
    condition (tosses == replicate 10 Head)
    nextToss <- coin
    return nextToss
   where
    fair   = choose (1/2) Head Tail
    biased = certainly Head