What is the probability that the student knew the answer to at least one of the two questions? A student takes a true-false examination containing 20
questions. On looking at the examination the student and that he
knows the answer to 10 of the questions which he proceeds to answer
correctly. He then randomly answers the remaining 10 questions. The
instructor selects 2 of the questions at random and  that the students answered both questions correctly.
What is the probability that the student knew the answer to at least one of the two questions?

When complete, the following table shows the probabilities of getting $2$, $1$, or $0$ of the two questions right depending on the kinds of questions. $K$ refers to a question with a known answer and $U$ to a question with an unknown answer; the first letter is for the first question graded, and the second is for the second question graded. I’ve omitted most of the entries, leaving them for you to compute.
$$\begin{array}{c|cc}
&2&1&0\\ \hline
KK&1&0&0\\
KU&a&&0\\
UK&b&&0\\
UU&c&&1/4
\end{array}$$
The student got both questions right, for a score of $2$, so the entries that you need are those in the first column: you need to determine what fraction of the total $1+a+b+c$ is contributed by the $KK$ case.
