[Math] conditional probability that both answers are correct given that at least one is correct

Question

I came across this question whiles doing conditional probability and need a vivid explanation. A person answers each of two multiple choice questions at random. if there are four possible choices on each of the question, what is the conditional probability that both answers are correct given that at least one is correct

Answer 1

Call the questions A and B. Given that at least one of these questions was answered correctly, we must be in one of the following seven situations:

$$\begin{array}{c|c} \text{Question A}&\text{Question B}\\ \hline \text{right answer}&\text{right answer}\\ \text{right answer}&\text{first wrong answer}\\ \text{right answer}&\text{second wrong answer}\\ \text{right answer}&\text{third wrong answer}\\ \text{first wrong answer}&\text{right answer}\\ \text{second wrong answer}&\text{right answer}\\ \text{third wrong answer}&\text{right answer} \end{array}$$

These seven situations are equally likely, and only one of them has both answers correct, so the desired conditional probability is $\frac17$.

(The other nine possible outcomes involve combining one of the three wrong answers to Question A with one of the three wrong answers to Question B; this can be done in $3\cdot3=9$ different ways.)