Great expectations

            Last week I finally managed to find out the probabilities of completing a game of Chutes and Ladders in various numbers of moves, so can we finally (mathematically) answer the nagging question, what is the expected number of moves required for a player to complete a game of Chutes and Ladders?

            When we determined this value using the computer simulation, we had actual data to work with: the total number of games played and the number of moves each game lasted for. We divided the latter number by the former number and out popped the average. Can we come to the same result with just some theoretical probabilities? 

            Well, we can get pretty close. To reiterate, when we're talking about the average number of moves per game, we mean the total number of moves divided by the total number of games:

            We can then break down the "total number of moves" as follows:

            Note that the numerator continues on infinitely since there is no theoretical limit to the maximum number of moves a game can last for.

            Now separating each term in the numerator into an individual fraction and simplifying...

            This is an expression we can work with, since we found the exact probabilities of completing the game in various numbers of move using the transition matrix. There is still one problem though. As already mentioned, this expression requires us to know the probability of completing the game in every possible number of moves, which is a theoretical impossibility since there are infinitely many. However, we can still make a good approximation based on a large number of moves. As you can see from the previous weeks' graphs, the probability of finishing a game in a triple-digit number of moves or more is very slim, so we aren't hurt too much if we leave a lot of these larger values out. Using the probabilities of completing the game in 1,2,3,...,300 moves, I got an approximate average of about 39.21529. The estimated average of completing the game from the computer simulation was also around 39.2, so not bad!

            While that much precision is probably good enough for Chutes and Ladders, there might be other scenarios where more precision is needed, so it's still important to know whether there's a way to determine the exact number of expected moves. It turns out there is a way to do this using (you guessed it!) matrices.

            This will probably be the most obscure part of this project to date (and everything else hasn't exactly been a skip through the park), and since I'm already nearing the 500 word limit here, I think I'll just push off that explanation into the next entry. Stay tuned.