I have coded the problem in R using ‘procedural’ code and ‘vectorised’ code with a million replicates. My ‘procedural’ version is:

  win.by.switch <- 0
  win.by.stick <- 0
  for(replicate in 1:1e6)
    {
    prize <- floor(runif(1) * 3 + 1)
    pick <- floor(runif(1) * 3 + 1)
    if(pick == prize) {
      win.by.stick <- win.by.stick + 1
      } else {
      win.by.switch <- win.by.switch + 1
      }
    }
win.by.stick
win.by.switch

This gives:

  > win.by.stick
  [1] 333675
  > win.by.switch
  [1] 666325

After running for about 18.6 seconds.

My ‘vectorised’ version is:

  prize <- floor(runif(1e6) * 3 + 1)
  pick <- floor(runif(1e6) * 3 + 1)
  table(ifelse(pick == prize, "STICK", "SWITCH"))

This gives:

   STICK SWITCH
  333339 666661

After running for 1.9 seconds.

The ‘vectorised’ version could be condensed to a one-liner:

 table(ifelse(floor(runif(1e6) * 3 + 1) == floor(runif(1e6) * 3 + 1), "STICK", "SWITCH"))

This gives:

  STICK SWITCH
  332902 667098 

After running for about 1.6 seconds but the code is far from self-documenting.

I apologise in advance if this is a stupid suggestion. Feel free to gloat if I have misunderstood the “Monty Hall” problem …. that would put me in some esteemed company.

And now I am a bit surprised that no one mentioned Excel. It’s certainly not the ideal platform for inquisitive computing, and it costs a lot of money, but in my line of work (engineering) everybody has a copy and knows how to use it. The spreadsheet itself is versatile enough, but if you know how to use the Basic interpreter you can do almost anything. Over the years I’ve used it quite a lot, to get answers to all kinds of problems.

I wanted to mention a programming language that has been very helpful to me. I’ll comment more about it in the future. It’s called J and is the latest incarnation of APL — it’s known as a matrix-oriented language (it is) and it’s also quite functional.

It’s worth a look, at http://www.jsoftware.com .

R is also a free alternative. It’s biggest disadvantage is its focus on statistical analysis but otherwise is an excellent program.

There’s also SciLab (French website) that is similar. Also free.

One issue with these programs is their focus on matrix/vector formulation. This is an advantage in making matrix or vector computation look simple but does require some rethinking of how one organizes one’s data. But worth the effort.

Personally, I use C and (more and more rarely) FORTRAN to do numerical work. Especially with things like Numerical Recipes available it’s easy to assemble a main program and calls to a few subroutines to do many quick problems like simple Monte Carlos. I did the Monte Hall problem in about 20 minutes when it first came out to “prove” it to our lunch group of physics PhD’s. Especially for more computationally intense problems the speed difference between a compiled language and the interpreted ones is significant.

(I didn’t post it three weeks ago because it barely existed then. Perhaps it’s still too crude to mention.)