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 `(31)/(32)` , then p lies in the interval : (1) `(1/2,3/4]` (2) `(3/4,(11)/(12)]` (3) `[0,1/2]` (4) `((11)/(12),1]`

Options

A. `((11)/(12),1]`
B. `((1)/(2),(3)/(4)]`
C. `((3)/(4),(11)/(12)]`
D. `[0,(1)/(2)]`

Explanation

To solve the problem step by step, we need to analyze the given conditions and apply the concepts of probability.

### Step 1: Understanding the Problem
We have 5 independent Bernoulli trials, each with a probability of success \( p \). We need to find the interval for \( p \) such that the probability of at least one failure is greater than or equal to \( \frac{31}{32} \).

### Step 2: Probability of At Least One Failure
The probability of at least one failure can be expressed as:
\[ P(\text{at least one failure}) = 1 - P(\text{no failures}) \]
The probability of no failures (i.e., all trials are successes) in 5 trials is given by:
\[ P(\text{no failures}) = p^5 \]
Thus, we can rewrite the equation:
\[ 1 - p^5 \geq \frac{31}{32} \]

### Step 3: Rearranging the Inequality
Rearranging the inequality gives us:
\[ p^5 \leq 1 - \frac{31}{32} \]
Calculating the right-hand side:
\[ 1 - \frac{31}{32} = \frac{1}{32} \]
So we have:
\[ p^5 \leq \frac{1}{32} \]

### Step 4: Taking the Fifth Root
To find \( p \), we take the fifth root of both sides:
\[ p \leq \left(\frac{1}{32}\right)^{\frac{1}{5}} = \frac{1}{2} \]
This means:
\[ p \leq \frac{1}{2} \]

### Step 5: Considering the Valid Range for \( p \)
Since \( p \) represents a probability, it must also satisfy:
\[ 0 \leq p \leq 1 \]
Combining this with our previous result, we conclude:
\[ 0 \leq p \leq \frac{1}{2} \]

### Step 6: Final Interval for \( p \)
Thus, the interval for \( p \) is:
\[ [0, \frac{1}{2}] \]

### Conclusion
The correct answer is option (3) \([0, \frac{1}{2}]\).

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