How to define my moments of a fake dice?

Question

Say I have a dice.

Say I suspect that the dice is unfair with probability 75 % with probabilities $p_1, ..., p_6$ that are NOT all equal to 1/6.

However, with 25 % probability, the dice is actually fair and the probabilities are indeed 1/6.

I have two questions:

1. What is the mean and variance of this dice's outcomes?
2. Are there different ways of defining what I meant by "fake with 75 % probability" and do those different ways lead to different answers? For example, maybe the die is EITHER fake OR real, and I don't know which one it is .... that is one way to define it. OR maybe the die is BOTH fake AND real, and it switches between these two states every time I throw it. This definition is different, but does it give the same answer? Are there other definitions that make sense?

Answer 1

You'd have to calculate it based on $p_i$, for instance on the average one would do: $$E(X) = \sum_{k=1}^6 k \cdot \left(\frac{3}{4}p_1 + \frac{1}{4}\cdot\frac{1}{6}\right) = \frac{3}{4}E(X|\text{fake}) + \frac{1}{4}E(X|\text{real}) $$ The equivalent way to see this question that would more intuitively work is saying you randomly choose a dice out of 4, 3 of which are identical but unfair with probabilities $p_i$, and $1$ is a fair dice, if the chance of an unfair dice were lower one could estimate the mean and variance well enough, but with a $\frac{3} {4}$ that a die can go from rolling straight sixes to rolling straight 1 means the range there is too big to get anything meaningful, in fact: $$\frac{13}{8}\le E(X)\le \frac{43}{8}$$

Answer 2

Regarding your second question, you can get different answers depending on what you mean by "unfair with 75% probability". In most cases, you will have dice with fixed characteristics that are either fair or unfair, with some fixed probability distribution over the faces. If the die is either fake or real, the expected value of the die is either 3.5 (if it's real) or whatever the expected value of the biased die is. Regardless of how much you believe the die is fair or biased, the die's actual expected value must be one of two values, and not the average of both.

If you did have a die that could change characteristics, the expected value would be the weighted average of the fair and biased expected values. The expected value of the die in this case is based on real, physical characteristics of the die, and not your own beliefs about the die's characteristics.

You could alternatively look at this as a process in which you randomly select a die from 3 biased and 1 fair die, and then roll it. In this case, the expected value of the process is again the weighted average of the fair/unfair expected values. But if you only have one die with immutable characteristics, its expected value is determined only by those characteristics and not how likely you think it is that the die is fair. The die actually is fair or not, and has some fixed, true expected value which is unaffected by how strongly you believe the die is fair.