Smallest trial count for an at-least-one threshold

A synthetic sensor gives an independent reading at each time point

On each reading, the probability of a reading in the stated range is \(\frac{3}{4}\), and the probability of a reading outside the stated range is \(\frac{1}{4}\). The trials are independent

Let \(X\) be the number of successes in \(n\) independent readings. The required probability threshold is \(0.9960\)

Use the relation \(\Pr(X\ge 1)=1-(1-p)^n\) for the probability of at least one success

Find the least value of n for which the probability of at least one success is greater than 0.9960