Dice randomness
We were playing Settlers of Catan the other day. If you don't know it, you should immediately go out and buy it, because it is a fantastic game... one of the best strategy / board games ever. But back to my topic... The game relies on die randomness. From basic math / probability, we know that when rolling two dice, the sum of 7 has the highest probability of getting rolled. Next, the sums of 6 and 8 would have the next best chance. More importantly, the 6 and the 8 would have equal chance of getting rolled. Then 5 and 9 and so on. This is easily shown with a table (sample space) displaying all the possible outcomes of rolling two dice and then counting the number of possible outcomes of getting a sum of 7, sum of 6, sum of 8 etc.
But when we were playing our board game, it seemed that these random rules didn't apply all that well. The higher numbers seemed to come up way more than the lower ones. For instance 9 came up way more than 5. First we thought that our dies are not well balanced, so we traded them from a set from Monopoly, but the same trend continued. Unhappy with this (I lost the game because I chose lower numbers in Settlers and they hardly came up) I decided to look into this more.
First of all I have a hypothesis. The high numbers come up more because the low number sides weigh more (less dots: less holes - more material left), so they tend to go to the bottom. This makes the higher numbers come up more. Since the 1 and the 6 are opposites of each other on a standard die, the six should come up more than the 1 as the 6 side weighs less -right?.
I looked it up on the internet, and it seems that the regular board game dice are not really balanced. The rounded edges make them more likely to be bias to one side more than the other. The true casino dice have pointy corners and straight edges. Also the amount taken out by the dot is filled with another material to compensate for the loss of mass. This way, the casino dice are as perfectly unbiased as you can get. But not so for the standard board game dice.
Also, online, I didn't find a good article or good experimental results to test my hypothesis (regarding the loss of mass on a higher numbered side). Therefore, being the inquisitive mind I am, I had to test it out myself:
I tested four regular board game dice. Rolling each 100 times, I checked whether they rolled each number about the same amount of times. This is a really standard and boring test, but I did it because I couldn't believe that game - how could the high numbers be so consistently rolled more often.
The results:
For 100 rolls, the number of outcomes of each face
Die # #1 rolled #2 rolled #3 rolled #4 rolled #5 rolled #6 rolled Prob. of Rolling 4, 5, or 6 First 23 8 13 20 18 18 56% Second 11 18 16 16 17 22 55% Third 14 8 14 19 25 20 64% Fourth 9 21 13 21 21 10 52%
All four experiments on four different dice show that it is more likely to get a higher number(4, 5, or 6) than a lower one(1, 2, or 3). These results support my hypothesis. Of course way more experiments have to be done before this can become something significant, but just from this small experiment I will definitely choose high numbers (as opposed to low) in Settlers of Catan, next time I play.
Even though this experiment was very trivial, I think this could be a cool lesson in probability. Even for the high end students, the math wizzes, it could be a good lesson in humility, as well as demonstrating to them that even the die is not as random as we might think! A good place to have a discussion in your math class.
Chi-squared test
A χ² test on the both the first and second sets of 100 observations reports that the difference from random is not statistically significant. There's a lesson in that.
I just wrote my final exam on Stats
I just wrote my final exam on Stats. Now I understand Chi Squared inside and out.
There's a few things that don't sit well with what you have said about how the results aren't significant. First of all, in Chi Squared the significance doesn't at all depend on the number of trials. So if I continued this experiment for 1000 times, and still got similar results, Chi Squared would still not give me a significant result - even though my results should be averaging to a random sample, but their not.
Secondly, chi squared compares values to expected values. In this case, if I assumed the expected values were not random but skewed towards the higher numbers coming up more often (as I hypothesized) then I could state there is significance.
Anyway, I think my experiment holds, even after considering the Chi Squared Test (as you pointed out).
Chi Squared Is Not Good
Chi-Squared test is not very sensitive. Both the first and second sets of 100 observations are not signifcantly different from a "rigged dice" in which rolling a 4,5, or 6 was 50% higher than rolling a 1,2, or 3.
If I were a casino owner I would certainly be able to create a very profitable game with dice like that!
With this many degrees of freedom we need a lot more samples to detect even large imbalances.
Just eyeballing it, the dice do seem heavy towards the high numbers. Probably the more dots the more material was drilled out of the face and the lighter it is.
get a bucket...
I work in the gaming industry and have some experience in testing "loaded" die. The chi-square analysis can reveal anomolies when you have a large sample size, like when your expected value is at least 200. So at a minimum you would need 1,200 results. I would recommend at least 12,000 results. As you can see, this would be impractical.
The drilling of the "pips", or dots, would not remove enough weight to cause any statistically significant change in weight from one side to another.
The best way to check for "loaded" die, throw it into a bucket of water. I'm not an expert on bouyancy or physics, but I know that if you throw a loaded die into a bucket of water, the heavy side will land on the bottom almost every time. And if the die is severely weighted, it will land heavy side down every time.
That's very cool. I will
That's very cool. I will have to try it out with my dice.
Interesting...
I once knew about the χ² test. It's all gone now, somewhere hidden in a far away corner of my brain, where I haven't looked for ages. (I have to relearn it again, so I don't look like a fool next time.) Thanks for the insight.
There is definitely a lesson in that... for me :)