Consider 5 independent Bernoulli trials each with probability of success p. If the probability of at least one failure is greater than or equal to 31/32, then p lies in the interval?

p (probability of a success)
Probability of at least one failure = 1 - (Probability of no failure)
For 5 events, probability of no failure = p⁵
Hence, 1 − p⁵ ≥ 31/32 → p⁵ ≤ (1/32) = (1/2)⁵ → p ≤ 1/2
But p ≥ 0
So, P lies in the interval [0, 1/2].