Assume you write a multiple-choice exam that consists of 100 questions. For each question, 4 options are given, one of which is the correct one. If you answer each of the 100 questions by choosing an answer uniformly at random, what is the probability that you have exactly one correct answer?
a) $\frac{100}{4^{100}}$
b) $\frac{3^{99}}{4^{100}}$
c) $\frac{100 + 3^{99}}{4^{100}}$
d) $\frac{100 \cdot 3^{99}}{4^{100}}$
Solution: 2014 Fall Midterm - 17
Author: Michiel SmidQuestion
Solution
We can expand it. First, we pick 1 out of the 100 questions. There are 100 possibilities.
Then, the probability that we got it right is $(1/4)$
Then, the probability that the second question we do is wrong is $(3/4)$
Then, the probability that the third question we do is wrong is $(3/4)$
We can write this as:
$100 \cdot \frac{1}{4} \cdot \frac{3}{4} \cdot \frac{3}{4} \cdot … \cdot \frac{3}{4}$
$=100 \cdot \frac{1}{4} \cdot \frac{3^{99}}{4^{99}}$
$=100 \cdot \frac{3^{99}}{4 \cdot 4^{99}}$
$=100 \cdot \frac{3^{99}}{4^{100}}$