Question

A multiple choice examination has 5 questions. Each question has three alternative answers of which exactly one is correct. The probability that a student will get at least one correct answer is

• A. \(\dfrac{80}{243}\)
• B. \(\dfrac{32}{243}\)
• C. \(\dfrac{163}{243}\)
• D. \(\dfrac{211}{243}\)

Hint:

“At least one” probability is easiest using complement: \(1 - P(\text{none})\).

Answer 1

The Correct Option is D

Step 1: Find the probability of getting a question wrong.
Each question has 3 options, and only 1 is correct. So, probability of correct answer on one question: \[ P(\text{correct}) = \frac{1}{3} \] Probability of wrong answer on one question: \[ P(\text{wrong}) = 1 - \frac{1}{3} = \frac{2}{3} \]
Step 2: Find the probability of getting all 5 questions wrong.
Assuming the student guesses independently for each question: \[ P(\text{all wrong}) = \left(\frac{2}{3}\right)^5 = \frac{32}{243} \]
Step 3: Use complement to find at least one correct.
\[ P(\text{at least one correct}) = 1 - P(\text{all wrong}) \] \[ = 1 - \frac{32}{243} = \frac{243 - 32}{243} = \frac{211}{243} \]