Is a dice more random than a coin?

Question

My roommate and I have gotten into an argument. I presented him with the question is a dice, with half of the sides numbered 1 and the other half numbered 2, more random than just flipping a coin? Statistically we agree that the chance is 1/2 either way. But I am arguing that a dice is better as a random generating object then the coin. My idea is that if both the coin and the dice were thrown, the coin would land and be finished "rolling" before the dice was done generating its number. With more computation time, the dice would be more random. My question then becomes, is a dice or any other object just better at randomly generating numbers? Would a person have a much easier time creating a random event with one object over another?

Answer 1

Randomness of these objects are due to the chaotic nature of their motion, especially when bouncing on a surface. Chaotic motion amplifies the initial conditions exponentially, so that even thermal noise could make a change to the outcomes.

Anyway, I would not be surprised that a well trained player, in repeatable conditions and able to choose the initial position could obtain a bias in the drawings. (Especially if catching the coin in hand, or throwing the dice on a short distance.)

But I have no insight on which could be most easily tricked.

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It is believed that quantum effects can generate perfectly unpredictable events. Note that modern Intel Processors include a RNG based on such effects.

Answer 2

Unless you specify in mathematical terms what you mean by "random" and "better" and "easier time generating," and unless you are talking about an idealized die and idealized coin, then your question is not really about mathematics.

Here are some ways in which we might be able to formulate your question mathematically:

1. If we specify the geometry and mass of the coin and the die, and for a fixed velocity choose a random angle and orientation to throw them, on average, what is the time for them to "settle?" This is a physics question and depends on a number of additional model parameters such as the proportion of energy lost per collision.

2. Assuming the coin and die each have a known and idealized probability distribution, and ignoring the amount of time to generate realizations from each, would a third observer who is furnished with samples of arbitrarily large but equal size from each object but not the information about which object generated which sample, be able to distinguish the two? This is a mathematical question.