Expected correct answers in a test

Text

A test consists of $20$ multiple-choice questions, each with $4$ options of which only one is correct. A student answers completely at random to all questions. What is the expected number of correct answers and its variance?

Solution

The number $X$ of correct answers is binomial $X\sim B(n;\,p)$ with $n=20$ and $p=\frac14$ (probability of guessing at random). For the binomial: $E(X) = np = 20\cdot\frac14 = \boxed{5},$ $\text{Var}(X) = np(1-p) = 20\cdot\frac14\cdot\frac34 = \boxed{3.75}.$