When a certain biased die is rolled, a particular face occurs with probability \(\frac{1}{6}-\mathrm{x}\) and its opposite face occurs with probability \(\frac{1}{6}+\mathrm{x}\). All other faces occur with probability \(\frac{1}{6}\). Note that opposite faces sum to \(7\) in any die. If \(0\,<\,x\,<\,\frac{1}{6}\), and the probability of obtaining total \(\mathrm{sum}=7\), when such a die is rolled twice, is \(\frac{13}{96}\), then the value of \(x\) is:

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.