## A loaded die study

I got a set of handcrafted dice for Xmas. They look like this:

They are wooden, the shapes are obviously imperfect, and the dots are made of metal.

I immediately suspected that they can’t be well balanced and decided to conduct a little study, that turned out to be **fun and instructive**.

The first experiment I did was dropping the die into water. It appeared that the die is lighter than water and it **always floats the 1 side up and the 6 side down**:

*[Side quest: A perfect cube with the density of exactly 1/2 of the density of water floats in water. Will it float side-up, edge-up or vertex-up? This problem is too tough for me, and I don’t have a solution.]*

Anyway, after the float test it was evident to me that the die is biased in the 1- 6 direction. But by how much is this imbalance affecting the outcome of rolling the die on the table?

My next test was the Roll Test. **I rolled the die 121 times. **Why 121? I wanted a number that is close to 100, close to a multiple of 6 and close to a perfect square. That is because I was pretty sure I’d be able to compute the sigma on a napkin and that would be it. Both 120 and 121 are good numbers. But after I did 120 rolls I thought, why don’t I do one more.

Here are the results of my 121 rolls:

1 | 2 | 3 | 4 | 5 | 6 |

27 | 19 | 19 | 20 | 21 | 15 |

Right, 1 and 6 are obvious outliers… ??? … But **they are within the 2-sigma range**… But I’m more than confident in my alternative hypothesis! Like a true researcher, I’m going to find a way to confirm it!

So, do I do another 100 rolls? No way, that would be no fun! I’m going to pretend that I’m not dealing with a stupid die, but rather with a particle accelerator, and that I’m over the budget, so this sample is all I have. I’ll do various stats tests on my sample, and I’m going to find one that confirms that this stupid die is loaded!

Sadly, both Chi-square and Kolmogorov-Smirnoff yield p-values about 0.5 that is 10 times greater than what I need in order to reject the hypothesis that the die is fair.

But I’m not giving up. Why was I doing all those tests that attempted to **refute the null-hypothesis without having any information about my alternative hypothesis: that the die is biased specifically towards 1 and specifically against 6**. And why was I doing all those old-fashioned tests at all? It’s not 19th century and I have a very capable computer at my disposal. I can simulate whatever I want.

Results:

Alternative Hypothesis (out of 121 rolls) |
p-value |

At least 1 number appears more that 26 times | 0.37 |

At least 1 number appears more that 26 times AND at least 1 number appears less that 16 times | 0.3 |

Exactly 1 number appears more that 26 times AND exactly 1 number appears less that 16 times | 0.2 |

The number ONE appears more that 26 times AND the number SIX appears less that 16 times | 0.01 |

So I guess I **could** now conclude that the die is biased specifically towards 1 and specifically against 6. But **should** I?

The next step should be checking how significant this bias is for some actual game. I’d need to simulate a game with a realistic bet and see how much money can be won using this die within a realistic timeframe.

I used R for the simulations. Below is the piece of code that does that. **Happy New Year!**

###### n = 121

###### isLoaded = function(freqs)

###### {

###### # H_alt

###### #sum(freqs > 26) == 1 & sum(freqs < 16) == 1

###### freqs[1] > 26 & freqs[6] < 16

###### }

###### trials = 10000

###### succ = 0

###### for(i in 1:trials){

###### # do n rolls

###### sim = sample(1:6, n, replace=TRUE)

###### # freqs

###### t = table(sim)

###### if (isLoaded(t))

###### succ=succ+1

###### }

###### succ/trials

If you hate loops in R, here is a loopsless version that does the same:

**s=replicate(trials, table(sample(1:6, size=n, replace=T))) sum(apply(s, 2, isLoaded))/trials**

This version has the same performance as the loop, needs more memory and hard to read. The only benefit is that you can compute your sample space (s) once and then quickly test various things on it.

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