Mahjong meets the Law of Large Numbers

One of the topics we cover in my statistics class is the Law of Large Numbers.  Basically, this law says that a sample statistic will approach its respective population parameter as the number of trials you perform goes to infinity.

For example, if you toss a fair coin 10 times, you might get any number of heads or tails between 0 and 10, even though the odds of getting either outcome is 50%.  If you toss that same coin 20 times, you still might not have half-and-half heads and tails.  But as you toss the coin even more times, say 100 times and then 1000 times and then 10000 times and so on, the proportion of heads (or tails, depending on what you are counting) you get will get closer and closer to 50%.

Occasionally I play a mental game to help my mind relax.  Lately I’ve rediscovered a game I haven’t played in years — Mahjong.  It’s a sort of Chinese solitaire game in which 144 tiles are stacked in a set pattern, and you win the game by matching all the tiles.  However, only tiles with no neighboring tile on the left or right are free for matching.  It’s simple and yet highly addictive in its challenge.

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As I played Mahjong about a month ago, I wondered what the odds of winning this game are.  Curious to know the answer to that question, I determined to use the Law of Large Numbers to find out.

I started keeping track of my games on 22 February, and to date I’ve played 669 games.  At the end of each of these games — of which I’ve won 117, or 17.49% — I count the number of tiles remaining.  With this tally I can answer questions like “Does my performance improve with time?”  I can do this in two ways: First, I can take the proportion of wins in the first 100 games and compare it with the same proportion for the last 100 games.  I can also perform a hypothesis test to compare the mean number of times left at the end of the game for the first 100 games with the same number for the last 100 games.

To use the Law of Large Numbers, I really need to conduct at least 1000 trials and preferably much more.  Admittedly, 669 isn’t that far away from 1000, but I’ll come back and update this blog with a future entry to report my progress as well as my findings when I’m done.

As an engineer, I tend to focus on practicality and what is actually useful for some purpose.  This question has no purpose; I pursue it because I’m just plain curious.  And it doesn’t take inordinate amounts of time.  I play a few games here and a few there, which I will do anyway to give my mind a break from all the thinking I do throughout the day.  I just need to remember to count the number of tiles at the end and record them in my spreadsheet for later analysis.

Stay tuned for updates.  I’ll report on how my curiosity experiment unfolds.