Cards numbered 1 to 200: A card is drawn uniformly at random. What is the probability the number on it is a perfect cube?

Introduction / Context:We need the density of perfect cubes among the integers 1 through 200. Recognize that cubes grow quickly, so only a few lie within this range.

Given Data / Assumptions:Numbers are 1,2,3,…,200 with equal probability; perfect cubes are n^3 for integer n ≥ 1.

Concept / Approach:Find the largest n with n^3 ≤ 200. Then probability = (count of cubes)/200.

Step-by-Step Solution:

Verification / Alternative check:Use floor of cube root: ⌊∛200⌋ = 5 ⇒ count = 5; divide by 200.

Why Other Options Are Wrong:3/40 and 7/40 miscount; 19/40 is not plausible given rarity of cubes.

Common Pitfalls:Including 216 (6^3) erroneously or forgetting 1 is a cube.

Final Answer:1/40