Guessing a quiz

Idea. Each guess is right with probability $\frac{1}{3}$, independently, so the number of correct answers is binomial.

Step 1: exactly 5. ${\left(\frac{1}{3}\right)}^5=\frac{1}{243}$.

Step 2: exactly 4. Choose the wrong question in 5 ways: $$5\cdot {\left(\frac{1}{3}\right)}^4\cdot \frac{2}{3}=\frac{10}{243}.$$

Step 3. $$\frac{10}{243}+\frac{1}{243}=\frac{11}{243}\approx \mathrm{0.045.}$$