Originally Posted by

**BeeCuz**
Please start from the premise that no errors have been committed, thank you.

The problem states ranges of dice rolls (confidence intervals) and the associated confidence level (probability) of the outcome being a member of the range. What I'm interested in is how to calculate the confidence interval for the associated probability.

For example, given the associated probability 10/36, what is the range of dice rolls that meet that specification? I know the answer beforehand, but that doesn't resolve how to calculate it.

The answer by the way is the range of values [2..9]

Let's forget the terminology, which you are still using inaccurately. That doesn't matter. What I think you want is this:

Given a pair of 6-sided dice and an interval [m, n], that is, 2 <= m <= x <= n <= 12, we can determine the probability that the sum of the dice, x, is in that interval, P(m <= x <= n). Is it possible to find an interval for which the probability will be any given number? That is, given p, we want to find numbers m and n such that P(m <= x <= n) = p.

The question still needs further clarification, though. There are infinitely many possible values you could choose for p (any real number between 0 and 1); but any probability we produce will be a fraction with denominator 36. Are we assuming that p = k/36 for some integer k? Without that, it would be impossible for most values of p.

Secondly, even with this restriction, there may be no interval that gives exactly p; or there might be more than one. Are you willing to accept that as the answer for some p, or do you want an interval that yields the closest possible probability to the given p? Your phrasing, "__the__ confidence interval for the associated probability", assumes that there is always one and only one answer, which is not going to be true.

By the way, for your example, I think you meant 30/36, as in the original question, not 10/36 as you said here. If not, then I'm doing something wrong.

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