Question

A student writes an exam with 8 true/false questions. He passes if he answers at least 6 correctly. Find the probability that he fails.

• A. \( \frac{37}{256} \)
• B. \( \frac{19}{256} \)
• C. \( \frac{119}{256} \)
• D. \( \frac{219}{256} \)

Hint:

Use the binomial probability distribution for problems involving multiple independent events with two outcomes.

Answer 1

The Correct Option is D

Step 1: Defining the Probability Distribution
Each question has 2 choices (true/false), so probability of correct answer: \[ P(\text{correct}) = \frac{1}{2} \] Step 2: Binomial Probability Calculation
The probability of exactly \( k \) correct answers follows: \[ P(X = k) = \binom{8}{k} \left( \frac{1}{2} \right)^8 \] Total probability of failing (less than 6 correct answers): \[ P(X \leq 5) = P(0) + P(1) + P(2) + P(3) + P(4) + P(5) \] Computing: \[ P(X \leq 5) = \frac{219}{256} \] Thus, the correct answer is \( \frac{219}{256} \).