Rolling dice on hard surface vs soft surface

Question

So I was playing a game of risk with some friends and decided to roll a six-sided die on the floor, which was carpet, as I had more space so I wouldn't knock any game pieces over.

My friend argued that it was unfair for me to roll a die on the carpet if someone else was rolling on a hard surface such as a table. I asked why they thought this and they explained that the friction of the carpet would affect the way the die rolls and thus bias the roll in my favor.

I don't know anything about probability or physics so I wasn't sure if this was true or not. I am wondering if the claim is true.

Answer 1

Just from a physics standpoint the kinetic friction is a factor in this situation as well as what is called the spring constant of the carpet vs. the floor. The spring constant is an intrinsic property of a spring and tells you how bouncy something is basically. Usually in physics they talk about actual springs with this property, but we can imagine a fiber of carpet as a tiny bouncy spring and assuming that when you walk on the carpet it returns to its original form after you pass over it (restorative force). Essentially there are many things at play here including the force at which the die are thrown. That will affect the restorative force. (Imaging jumping on a pogo-stick where you start to get real lit, and really push down on the spring so the restorative force flings you up.) The kinetic friction depends on both the coefficient of static friction (a intrinsic property of the carpet) and the normal force which is your mass of die times gravity. Also another thing at play here is impulse. In mechanics we learned that an impulse is a force divided by the amount of time. If you have a large force over a short time and conversely a large amount of time with a small force, you can create different impulses(and to confuse you more you can get the same impulse in two situations with different forces and times, just gotta do the math right). Thats why when you drop an egg on a carpet vs. dropping it on a hard floor, you get a different result. So throwing die on a carpet vs. floor would get my physics brain buzzin' because I would be surely debating that the material the die is thrown on can affect how it lands.

Statistics and combinatorics of how the die will roll is a whole other disgusting beast that I would never want to tackle. I'd just roll with the fact that the carpet vs. floor can give rise to some different results.

As far as it being in your favor...if you were skilled enough and calculated the amounts of force, the spring constant, etc required to roll onto a specific face of the dice, then yeh ok...you cheatin! The roll will slow but who says thats in your favor unless you are rolling it in a way where people can obviously see your making it roll two times so that the frictional force can stop it faster than usual. :)

Answer 2

The carpet will cause the die to turn fewer times than the hard floor. Claiming it will bias the rolls is much harder. The claim would be that someone could be skillful enough to throw the die in a manner that the rolls would come up more often one way than another. At the extreme, let the surface be very sticky so that the die stops instantly on touching the surface (then magically settles so one face is on top). Now if you can throw it without spinning you can choose the face that ends on top. If the carpet steals energy from the die you might be able to learn to throw it so it would roll three quarter turns before stopping (maybe not all the time, but more than other possibilities). You could then throw it with a bias. By comparison, if it turns twenty or so times on the hard floor, it is harder to believe someone could throw accurately enough to have a bias in the results.

In the past I have seen a similar controversy when throwing into the reasonably hard box top. Sometimes people would throw the dice in a way that they skidded and did not roll (much). Should those be rerolled? If the roller didn't position the dice in his/her hand, maybe that is the randomness you need.

It's hard. You clearly do not keep enough statistics to really know. In games where a high roll is always good, if a skillful roller knew you were keeping count he/she could choose the times to get bad rolls because they didn't matter much. In other games it is not clear what number is best, so it is even harder to collect a bunch of data an show it is outside expected variation.

Answer 3

So the entire randomness in this process comes from you guys shaking your hands and throwing the die. Now let's say you both shake your hands randomly without any adversarial intent then there could be no bias, because even if the carpet die does not role at all, and the board dice rolls $n$ times you added randomness when you shake the dice.

Now comes the fun part, let's say you guys are both allowed some skillful throw of a die and you gain information about the dynamics of the roll as the game proceeds. Then you could both, in principle eventually learn to throw at a perfect angle and velocity such that the number of your choice lands(the trajectory can be predicted given the initial velocity and position)

So now, how can there be bias? Case could be made that the lesser rolls of the die easier it is for you to learn the dynamics and reach the ideal case where you can always role the number you want to role. In that case, the one amongst you playing on the carpet will be able to adapt to the perfect throws quicker.