माना एक अभिनत सिक्के के लिए चित आने की प्रयिकता \(\frac{1}{4}\) है। इसे बार-बार उछाला जाता है जब तक कि चित प्राप्त न हो जाऐ। माना सिक्के को उछालने की आवश्यक संख्या \(\mathrm{N}\) है। यदि समीकरण \(64 \mathrm{x}^2+5 \mathrm{Nx}+1=0\) के वास्तविक हल न होने की प्रायिकता \(\frac{\mathrm{p}}{\mathrm{q}}\) है, जहाँ \(\mathrm{p}\) तथा \(\mathrm{q}\) असहभाज्य हैं, तो \(\mathrm{q}-\mathrm{p}\) बराबर है_______

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.