First movers advantage – Chutes and Ladders

I’ve always been fascinated by probability – the idea that if you understand how a system works, you can understand how likely a given outcome is. For example, since I know that there are 13 clubs in a deck of 52 cards, I know that I have a 1/4 chance of pulling a club at random.

This love of probability led me to purse a masters in statistics and it’s why I started my career as an actuary. But, it’s not outcomes that I am interested in, it’s the system that created the outcome. I am convinced that “systems thinking” is a skill that is crucial to be successful. That would make for a great blog post! For now I will just geek out on stats.

I was playing Chutes and Ladders with my oldest daughter when I decided I would try and understand the system of Chutes and Ladders and the probabilities associated with the game. You remember playing Chutes and Ladders right? Picture below:


It was during a particularly long game of C&L that I decided I would code the game into excel and run thousands of simulations to understand the distribution of results (ok, someone please send me an official geek card). The screenshot above is what the board looks like in excel. I was interested in answering the following questions:

  1. How many spins should a game take, on average (a reasonable question when you are in the middle of a twenty minute game of C&L)
  2. What’s the shortest number of spins that could win a game?
  3. How do the dynamics of #1 and #2 change if you add players?
  4. Most importantly: what advantage is there for going first – my daughter always goes first. Surely that counts for something, right?

For #1 and #3, the graph below shows the averages


For #2, the minimum number of spins is 7.

For #4, in a 4 player game player #1 is almost 8% more likely to win than player 4.

Maybe someone else can take up the mantel here and tell me the probability of winning, without hitting a chute or ladder.

In addition to answering some geeky stat questions, I also wanted to show my daughter that you can feed the computer some instructions (code) and have it do all sorts of magical things. She was interested, but mainly wanted to make sure that she wins the game.

Like I said above, I’m sure there is some deep meaning in this exercise – for me it was a fun exercise. I wanted to get the blog post out there – the deep meaning can come later.

p.s. if you are interested in seeing the excel file, drop me a line and I can send it over