Probability
Classical Probability
Grade 12
Question:
<p>In a multiple-choice question, there are four alternative answers of which one or more answers are correct. A candidate gets marks if he ticks all the correct answers. The candidate, being ignorant about the answers, decides to tick at random. How many attempts at least should he be allowed so that the probability of his getting marks in the question may exceed \(\dfrac{1}{4}\)?</p>
Step-by-Step Solution
Key Concept: There are 2^4 = 16 possible ways to tick answers (each option can be ticked or not). The candidate needs the probability of randomly selecting the exact correct subset to exceed 1/4, which requires comparing the number of attempts needed against the complementary probability of failure.
<p><strong>Step 1: Count total possible answers.</strong> Since each of 4 options can be either ticked or not ticked, there are 2^4 = 16 total possible ways to fill the answer (including ticking nothing).</p><p><strong>Step 2: Identify valid outcomes.</strong> Only ONE of these 16 ways corresponds to ticking all and only the correct answers. So probability of success in one attempt = 1/16.</p><p><strong>Step 3: Set up the inequality.</strong> If the candidate makes n attempts independently, the probability of getting marks at least once is: P(success in n attempts) = 1 - (1 - 1/16)^n = 1 - (15/16)^n</p><p><strong>Step 4: Solve for n.</strong> We need: 1 - (15/16)^n > 1/4</p><p>⟹ (15/16)^n < 3/4</p><p>Testing values:<br>• n = 1: (15/16)^1 = 0.9375 > 0.75 ✗<br>• n = 2: (15/16)^2 = 0.8789 > 0.75 ✗<br>• n = 3: (15/16)^3 = 0.8242 > 0.75 ✗<br>• n = 4: (15/16)^4 = 0.7725 > 0.75 ✗<br>• n = 5: (15/16)^5 = 0.7244 < 0.75 ✓</p><p>∴ Answer: <strong>5</strong></p><p><em>Note: The given answer of 4 appears inconsistent with standard probability calculation. Verification shows n = 5 is required for the probability to exceed 1/4.</em></p>
Correct Answer: 4