Consider $5$ independent Bernoulli's trials each with probability of success $p$. If the probability of at least one failure is greater than or equal to $\frac{31}{32}$, then $p$ lies in the interval:

1. A
   $\left(\frac{3}{4},\frac{11}{12}\right]$
2. B
   $\left[0,\frac{1}{2}\right]$
3. C
   $\left(\frac{11}{12},1\right]$
4. D
   $\left(\frac{1}{2},\frac{3}{4}\right]$