# Fairness of Cowry Shells as Dice

I just learned about the use of cowry shells as binary dice and find the idea enchanting. However, I'm concerned about the apparent "fairness" of the dice. Obviously, because there are multiple instances, they are not an equivalent to a single die with equal sides, (1d6 vs. 6*d2-1) as they'll have a curved probability distribution, rather than flat probability, and have an incredibly low chance of rolling 0. More than that though, it seems apparent that each shell would have subtle proportional variations depending on the lifestyle of the cowry it came from, and these would likely make each shell prefer one state or another, meaning each instance would be weighted. Also, cowry shells may just, in general, not be especially 50/50 in the first place the way coins are. (I'm OK with that, but the individual weighting bothers me) I can't seem to find any statistical examination of their balance or probability.

If I wanted to use cowry shells as dice in a home made game, what could I do to improve the apparent fairness of their application?

This has been a problem since the days when dice, hand-carved out of sheep's knucklebones, could never be considered mathematically fair. Historically, there have been two principal ways of improving a game:

• Make sure that a particular number is not always good or always bad. In Craps, rolling a 7 is good for the shooter on the first roll, but bad thereafter; a slight unevenness making 7 more probable is not likely to materially affect the chances of winning, which it might with a simpler rule.
• Where possible, make results comparative not absolute. To decide randomly who starts in a two-player game, it would be possible to roll two dice and choose based on even or odd; this is 50/50 if the dice are fair, but not otherwise, so it is rare. Each player rolling the dice and the higher result starting is common, giving even chances no matter what unevenness there may be.

I would also suggest using all the shells every roll if possible; it is annoying enough waiting for those players who carefully select the "high-rolling" or "low-rolling" dice each time, without the possibility that they might actually have a point.

Cowry shells have a roughly 30% chance of rolling a 0. Depending on the shell, that can get as low as 18.65% and as high as 39.11%. (At least, with the test subjects I had) It seems that the larger the shell is, the less likely it is to roll 1. Even between shells of similar size, there is significant variation. So, as predicted and advised, they are totally unfair. I feel sorry for those ancient gamblers.

To create the impression of fairness, I collected 24 shells of approximately equal size and put them in a black bag. Before rolling, players pick out six shells, blind and random. After rolling, they return the shells to the bag. So, although any given roll is weighted, the way it is weighted is unpredictable for the people involved, and the seemingly identical appearance, size, and shape of the shells reduces the chances of cheating by feeling for the best shells. For the players, it gives an impression of fairness without mandating theoretical fairness in implementation. (Which is impossible, given the item in question.) To reduce the impact of any given roll being weighted in some way, win/loss conditions are only determined after many rolls, with the exact number dictated by the plays made during the game. Because no single roll determines a victory, one cannot reasonably blame the shells for their failure.

• Interesting research! Your statement about theoretical fairness isn't quite right. It's possible to get a fair result from a biased coin: e.g. billthelizard.com/2009/09/… The downside is that you need more flips of the same coin (rolls of the same shell). Given enough rerolls you can simulate a completely fair die with a single shell. – tttppp Mar 9 '17 at 6:49
• What is 0 in this case. The open side? – user32849 May 18 '18 at 10:40
• That's an excellent question @user32849; I don't know what @jamalcolmson considers a 0, but I'm personally using a simple system of visual similarity of the side that ends up: the curved side resembles a 0, while the flat side has an opening which looks somewhat like number 1. :) – Berislav Lopac Jun 20 '18 at 22:06

I think that fairness in the game is more dependent upon everyone using the same cowry shells, and less on having a 50/50 probability. The game would not be fair if one player used coins, and another used cowry shells.

The question reminds me of playing Shagai. Shagai is the Mongolian word for an ankle bone (sheep to be specific). Mongolians have come up with many games which use shagai. One of these games is a "horse race" where the shagai are rolled like dice. There are 4 sides a rolled ankle bone will come up on, one of which is called horse. Each time a player rolls a horse they get to move one space, and the first to reach the finish wins. The probability of rolling a horse is not 1/4 but the game is fair because all players roll the same shagai.

Cowry shells would be perfectly fair, as long as everyone gets to use the same ones, and rolls the same number of them each time.

• The problem with that is with a dice that are not weighted evenly it can be used by some to their advantage to get the rolls they need while others may not be able to. – Joe W Apr 19 '17 at 0:17

They are nowhere near 50/50. I'm guessing the round part is much more likely to land down. So, unless you want a game that takes into consideration the lower probability, using such shells would be detrimental.

That said, the odds would be the same for everyone if everyone used the same set of shells. Maybe even let the players select from a few to introduce a new element to the game.