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 $$\dfrac{31}{32}$$, then $$p$$ lies in the interval:

The probability of getting a failure in a single Bernoulli trial is $$1-p$$ and the probability of getting a success is $$p$$.

For 5 independent Bernoulli trials, the probability that \emph{all} 5 trials are successful is given by the multiplication rule: $$P(\text{all 5 successes}) = p^{5}\,.$$

The probability of “at least one failure” is the complement of the above event, so

$$P(\text{at least one failure}) = 1 - p^{5}\,.$$

According to the condition in the question,

$$1 - p^{5} \;\ge\; \frac{31}{32}\,.\tag{1}$$

Rearrange inequality (1):

$$p^{5} \;\le\; 1 - \frac{31}{32} = \frac{1}{32}\,.$$

Write $$\frac{1}{32}$$ as a power of 2: $$\frac{1}{32}=2^{-5}\,.$$ Taking the positive 5th root on both sides yields

$$p \;\le\; (2^{-5})^{1/5} = 2^{-1} = \frac{1}{2}\,.$$

Since a probability cannot be negative, the complete interval for $$p$$ is

$$0 \;\le\; p \;\le\; \frac{1}{2}\,,$$

which in interval notation is $$[0,\frac{1}{2}]$$.

Thus $$p$$ lies in the interval $$\left[0, \dfrac{1}{2}\right]$$.

Option B which is: $$\left[0, \dfrac{1}{2}\right]$$