I'm investigating Sketchy Dice Inc. for purportedly selling unfair dice with the following probability distribution:
\begin{align} p(1)&=0.1 \\ p(2)&=0.3 \\ p(3)&= 0.2 \\ p(4)&= 0.15 \\ p(5)&= 0.15 \\ p(6)&= 0.1 \end{align}
In order to obfuscate their villainy, Sketchy Dice Inc. has removed the numbers from the faces of all the dice shortly before a police raid.
I was given one dice and asked to determine whether we can say it does not have the above distribution. Let's say you roll the dice 100 times. If the faces were labelled you could simply do a standard chi squared test using the statistic:
\begin{equation} \chi^2 = \frac{\left(O_i-E_i\right)^2}{E_i} \end{equation}
and compare to the chi squared distribution with 5 degrees of freedom. Here $E_i = 100 p(i)$ is the number of times we expect to roll $i$ given a dice made by Sketchy Dice Inc and $O_i$ is the number of times $i$ is actually rolled. However, with the faces scrubbed, what can we do? My first inclination would be to sort $E_i$ and $O_i$ before performing the calculation, but it is no longer clear that this loss of labeling information does not invalidate the test.
How does a lack of correct labeling information impact this statistical test, to what extent is it still valid, and if it is completely unworkable, is there any viable alternative? Let's say we aren't being paid to roll dice and don't want to rely on being able to produce more than one observation of 100 rolls.
Note: I know I could alternatively test if the dice is unfair in general, but that's not what I'm asking. The above distribution is particularly heinous and making such a die would merit a higher prison sentence than just making an unfair die.