How to reduce type I and II error for determining bias of loaded dice for reduced legal costs? Suppose you are running a casino and that you are responsible for ensuring that all the dice are fair to avoid lawsuits. In order to do this, you might take a mean of 1000 throws of each die and perform a hypothesis test [using the central limit theorem, CLT] to see whether they are likely biased.
The average cost of a lawsuit is $£240,000$, whilst the cost of a die is $£3$, so in order to minimise costs would you aim to have $240000 \,\beta = 3\,\alpha$, where $\beta$ is Type II error, A.K.A., false negative rate, and $\alpha$ is the significance level of the hypothesis test (and also the probability of a Type I error, A.K.A., false positive rate)?
Now, in order to find the optimal $\alpha$ value, one must know the value of $\beta$, something that can only be calculated from die outcomes (which is what we are testing for in the first place), so the optimal solution cannot be found. That being said, however, I'm sure scenarios like this arise rather often, so how are they usually dealt with?  
tldr: How would you find a threshold value for the mean of a die above (or below) which it should be considered biased whilst also keeping $240000\,\beta \text{ error} \approx 3\,\alpha \text{ error}$? 
Edit: It seems my choice of example is rather poor, as a die shouldn't even be tested for fairness with a location test. That being said, however, my question more concerns the tradeoff between Type I and Type II error than the fact that a die is being used.
 A: I'm not sure I'd approach the problem this way. 
First, you have two hypotheses: the die is fair or not fair.  If it is not fair, the distribution of throws will not be uniform (taking a mean of the throws is not an efficient way to measure this). Rather, I'd record the distribution of throws and after a few dozen, start calculating the empirical probability distributions (perhaps by chi-square or by way of binomial distribution depending on the game being played). 
Factoring in the cost of a lawsuit doesn't enter into the equation for me, because the die should be eliminated regardless as soon as you're certain it's not fair. That said, there are applications of the sequential ratio probability test (SPRT) that are geared towards this. E.g., detecting manufacturing anomalies based on a sequential sample of product. 
A: TLDR – the problem is underdefined
Aim
The details provided in the question can only lead to a relative balance of errors. Simply rearrange the equation and $ P(TypeII Error) ≈ 3P(TypeI Error)/240000$. 
This is not what is wanted or needed, rather it is to know what threshold for mean that would be used as a cut off. I.e. what effect size minimises the cost function of your situation. 
The cost of a typeI and typeII error have been defined, but not of the test to determine these. 
Why does this matter?
The effect size that you can reliably measure depends on the acceptable type I error and the acceptable type II error. Below is a typical sample size calculator ( Box 1 in https://academic.oup.com/ndt/article/25/5/1388/1842407, open access) (OP indicated a generic answer was preferred so I have used a formula for normal distributions as this s probably more widely used, the principles are the same for other distributions types)
$$n = 2*((a + b)^2δ^2)/(μ_1−μ_2)$$where a is the z multiplier to achieve the desired $\alpha$ level (1.96 for $\alpha$ = 0.05), while b is the equivalent for $\beta$. $\sigma$ is the standard deviation of the test result and $(\mu_1-\mu_2)$ is the effect sizem, we can simplify this to the threshold since an unbiased average would be 0 so we can drop $\mu_2$). $n$ in this specific example is the number of repeated trials to be carried out on an individual die. Rearrange this to solve for the effect size:
$$ \mu_1 = 2*((a + b)^2δ^2)/(n)$$
What does this equation tell us?
The issue of type I and type II error trade-off only applies when sample size and effect size are set to a constant.
You can push both $alpha$ and $beta$ down ever lower and detect the same effect size by increasing the number of repetitions. But repeating a test increases the cost.
Since your costs link type I and II errors you can replace $a+b$ with $Z_\alpha + Z_{3\alpha/240000\alpha}$ 
Then you will need to either:
1)  define an upper limit on what you are willing to pay for testing the dice, this will then leave you with two variables to solve for (effect size and alpha)
2)  solve for repetitions, alpha and effect size simultaneously
You either need to define even more limits up front or use some post-hoc decision criteria to decide which balance of the possible outcomes is best suited.
If this were a reflection of a real world example, you would want to do a more detailed cost/benefit assessment (non-comprehensive list). 


*

*The cost of repetitions for your measure of fairness (this will guide your sample number, if it is say £1 per 10 reps, then it will cost 33x the cost of the dice to do 1000 reps).

*The cost of failing to detect a bias (you give this as £240000 in you example for legal expenses, but in the real world the loss of earnings due to reputation damage and many others may be worth throwing in)

*The cost of rejecting a fair dice (given as £3 in this case, in more generic cases there may be other issues to consider, including revenue boost due to enhanced reputation)

*If you want estimates of absolute costs and risks you would need to define absolute values for how the scenario would be used in the real world. E.g. how many dice do you envisage deploying in the real world and how many times will each be used? 
A useful reference:
https://www.graphpad.com/guides/prism/7/statistics/stat_sample_size_for_which_values_o.htm?toc=0&printWindow
A: As I understand, the core part of the question is "how to choose $\alpha$ and/or $\beta$"?
Maybe it's me, that is (non-statistically) biased to interpret the question like this. But recently I had numerous discussions and thoughts about this very question. In a nutshell: there is no general rule, it depends on each and every case.
Textbook examples usually contain an outside element, e.g. what kind of error is "morally"/"socially"/"politically"/etc. more or less acceptable. In your case it might be "financially". Examples are: 


* In law/court, what is worse: that a criminal walks free (type A) or that an innocent person goes is sentenced (type B).


* In medicine, a test: gives a false positive -- a healthy person is tested as sick or a false negative -- a sick person tests as healthy.

In any case, the discussion becomes something entirely non-statistical, non-mathematic. In the somewhat unusual/contrived/complicated example the additional step involves the calculation of expectation values and the minimization of costs/risks in light of the interpretation.
I somewhat digress, but it's similar to the "risk management" in a famous movie, that is intended to offend: a car company calculates the cost of a call-back for repairs versus the cost of a lawsuit and ensuing damages for accidents that are caused by the particular failure. The second calculation is based entirely on statistics, but the decision rule surely isn't.
In the case of the casino, it's simply regulated. There are no lawsuits for "unfair" dice. And in the case of a truly unfair die, the plaintiff would have to prove it that it is the casinos' fault. In online casinos the inverse is true; there are departments for fraud detection.
