Another article about UB Dice statistics! Hurrah!!

This article was inspired by 3 things.

- Nick Williams’s article about UB dice. https://daedle.net/2020/05/25/ubdice/?fbclid=IwAR22OKMpv0yshUdfqqQHI9T3HCyIVXmFfTjoCAszFWEFJf7sy73_HZMW6fg
- A comment by Dan King during a commentary on a UB game in which he suggested that rolling a nerve check and then immediately re-rolling the nerve check due to inspiring leads to a higher than average chance of rolling the same numbers.
- My game against Moonraker Andy last night which came down to a turn 7 game-defining nerve test in which I rolled a 9, immediately re-rolled it and scored the same numbers again thus turning a draw into a very unlikely win.

I started by rolling a single dice a lot of times. It immediately became apparent that the number 5 was grossly under-represented in the results, but as I rolled more dice this corrected itself and, within the bounds of the statistical sample, I could see no long term bias. I recorded the numbers of repeat rolls and ended up with the highest chain being 5 consecutive 6s. There is, I think, about a one in 1,296 chance of rolling the same number 5 times in a row and given the size of sample used, rolling one chain of 5 is not unexpected.

These tables show how the numbers rolled on one D6 varied over time. Sample 2 used the data from Sample 1 and then added more rolls.

I then moved on to rolling 2 X D6s to replicate nerve checks to see how often the same combination of dice arrived consecutively, for example 1 and 4 followed by another 1 and 4. I rolled pairs of dice in quick succession to see if this had any bearing on the outcome. Interestingly, had I stopped after my first rolls, the statistics would suggest there is a 100% chance of rolling the same numbers as I rolled 3 and 4 immediately followed by 3 and 4.

In fact I rolled a much larger sample and was quite surprised by the results. Ten % of the successive rolls resulted in the same numbers appearing. This was split exactly between rolls made immediately and those not rolled immediately with 5% for each. This struck me as a much higher percentage than I would have expected. And so I did some maths (did I tell you that I had to resit my maths A level at school?).

If you have a blue dice and a red dice, the chance of rolling exactly the same number on each in the next roll is 1 in 36. But, UB dice does not let us know which of the 2 dice rolled scored which number and so the chance of repeating ant particular roll such as 3 and 4 is in fact 2 in 36 as you can roll 3 and 4 or 4 and 3. For doubles such as 1 and 1 (6 in 36 occasions), the chance remains one in 36.

The chance of repeating the same dice roll is therefore (I think):

[(6 x 1/36) + (30 x 2/36)] / 36 = 5.1%

So the theoretical chance of repeating the same double dice roll in the next turn is about 5%, or about one on 20, which is probably more than most people expect but is only half of what I observed in my trial. It is possible that the difference is due to the sample size and that a larger sample size might produce results that more closely match the theoretical result.

Conclusions:

- Individual dice rolls do not favour any particular number.
- The theoretical chance of rolling the same numbers in consecutive nerve tests is about 5%. This is probably more than players might expect.
- In the practical experiment, 10% of nerve test rerolls resulted in the same numbers. This is double the theoretical result but might be due to sample size issues.
- In the practical experiment, there was no difference in the chance of rolling the same number in a reroll whether the reroll was made immediately or if there was a delay.

Hope this helps!