*Bounty: 50*

*Bounty: 50*

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 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 £240000, whilst the cost of a die is £3, so in order to minimise costs you would aim to have $240000P(textrm{Type II Error}) = 3alpha$ where $alpha$ is the significance level of the hypothesis test (and also the probability of a type I error). The cost of testing the die may be ignored.

Now, in order to find the optimal $alpha$ value, one must know the value of $P(textrm{Type II Error})$, something that can only be calculated if the actual mean of the die (which is what we are testing for in the first place) is known, 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 $240000P(textrm{Type II Error}) approx 3P(textrm{Type I Error})$

**Edit:** It seems my choice of example is rather poor, as a die shouldn’t even be tested for fairness with a test like this. That being said, however, my question really concerns the tradeoff between Type I and Type II error, not the die in particular.