Step 1: Understanding the Concept:
This is a binomial probability problem. Each question is a Bernoulli trial with probability of success \( p = 1/4 \) and failure \( q = 3/4 \). We want to find the probability of 4 or 5 successes in 5 trials.
Step 2: Key Formula or Approach:
Binomial Distribution formula: \( P(X=r) = \binom{n}{r} p^r q^{n-r} \).
Required probability: \( P(X \geq 4) = P(X=4) + P(X=5) \).
Step 3: Detailed Explanation:
Given \( n=5 \), \( p = 1/4 \), and \( q = 3/4 \).
For \( X=4 \):
\[ P(X=4) = \binom{5}{4} (\frac{1}{4})^4 (\frac{3}{4})^1 = 5 \times \frac{1}{256} \times \frac{3}{4} = \frac{15}{1024} \]
For \( X=5 \):
\[ P(X=5) = \binom{5}{5} (\frac{1}{4})^5 (\frac{3}{4})^0 = 1 \times \frac{1}{1024} \times 1 = \frac{1}{1024} \]
Summing the probabilities:
\[ P(X \geq 4) = \frac{15}{1024} + \frac{1}{1024} = \frac{16}{1024} \]
Simplifying the fraction:
\[ \frac{16}{1024} = \frac{1}{64} = \frac{1}{4^3} \]
Step 4: Final Answer:
The probability is \( 1/4^3 \).