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

Introduction / Context:
This problem deals with Bernoulli trials and connects the probability of at least one failure with the underlying probability of success. It requires translating a condition on the probability of a compound event into an inequality involving p and then interpreting the solution as an interval. This is a standard application of the complement rule and basic algebra.

Given Data / Assumptions:

• There are 5 independent Bernoulli trials.
• Each trial has probability of success equal to p.
• Therefore probability of failure on a single trial is 1 minus p.
• The probability that there is at least one failure in the 5 trials is greater than or equal to 31/32.
• We need to determine the range of values of p that satisfy this condition.

Concept / Approach:
The key idea is to use the complement of at least one failure. The event at least one failure is the complement of the event no failures at all. No failures means all trials are successful. Because the trials are independent, the probability that all 5 are successful is p^5. Therefore, the probability of at least one failure is 1 minus p^5. We set this greater than or equal to 31/32 and solve the resulting inequality for p.

Step-by-Step Solution:
Let F be the event that there is at least one failure. P(F) = 1 - P(no failure). No failure means all 5 trials are successful, so P(no failure) = p^5. Given condition: P(F) greater than or equal to 31/32. So 1 - p^5 greater than or equal to 31/32. Rearrange: p^5 less than or equal to 1 - 31/32 = 1/32. Thus p^5 less than or equal to 1/32. Note that 32 = 2^5, so 1/32 = 1 / 2^5 = (1/2)^5. Therefore p^5 less than or equal to (1/2)^5 implies p less than or equal to 1/2, because p is between 0 and 1. So the allowed interval for p is 0 less than or equal to p less than or equal to 1/2.

Verification / Alternative check:
Check boundary values. If p equals 1/2, then p^5 equals (1/2)^5 which is 1/32. Then P(F) = 1 minus 1/32 = 31/32, which satisfies the given condition exactly. If p is smaller than 1/2, then p^5 is smaller than 1/32, so 1 minus p^5 is greater than 31/32, which still satisfies the condition. If p is greater than 1/2, then p^5 becomes larger than 1/32 and 1 minus p^5 becomes less than 31/32, which no longer meets the requirement. This confirms that the interval [0, 1/2] is correct.

Why Other Options Are Wrong:
(0, 1) is too wide because values greater than 1/2 do not satisfy the inequality. [1/2, 1] includes values where p is greater than 1/2, which lead to a probability of at least one failure less than 31/32. (1/2, 1] is even worse because it excludes the valid boundary value p equal to 1/2 and contains only values that violate the condition.

Common Pitfalls:
Learners sometimes confuse at least one failure with exactly one failure, which would require a binomial expansion instead of using a simple complement. Another common error is to forget that probabilities lie between 0 and 1 and to take an incorrect root when solving p^5 less than or equal to a fraction. Always apply the complement rule first for at least one type questions and then solve the resulting inequality carefully with attention to the allowed range of p.

Final Answer:
The probability condition is satisfied when p lies in the interval [0, 1/2].