Mathematics

Mathematics

Problem

A student answers 1010 multiple-choice questions by guessing. Each question has 44 options with exactly one correct answer. What is the probability that the student answers at least one question correctly? a) 1−(14)101-\left(\dfrac{1}{4}\right)^{10} b) 10(14)(34)910\left(\dfrac{1}{4}\right)\left(\dfrac{3}{4}\right)^{9} c) 1−(34)101-\left(\dfrac{3}{4}\right)^{10} d) (34)10\left(\dfrac{3}{4}\right)^{10}
Intuitive explanation

The key insight is that "at least one correct" is much easier to count using its complement: the probability of getting zero correct. Since each guess has a 14\frac{1}{4} chance of being right, it has a 34\frac{3}{4} chance of being wrong. Find the probability all 10 guesses fail, then subtract from 1 to get your answer.

Intuitive explanation

The key insight is that "at least one correct" is much easier to count using its complement: the probability of getting zero correct. Since each guess has a 14\frac{1}{4} chance of being right, it has a 34\frac{3}{4} chance of being wrong. Find the probability all 10 guesses fail, then subtract from 1 to get your answer.

Browse the problem library