# What is the probability that he answers $4$ correct answers of the first $5$ questions?

We have a test of $10$ questions. Suppose we have a student that answer $8$ correct answers. What is the probability that he answers $4$ correct answers of the first $5$ questions? There isn't any connection between the questions in the test.

As we already know that the student has answered correctly 8 questions, for him to answer $4$ correct answers in the $5$ first, he must have a mistake in the first $5$ ones, and another in the $5$ last ones. So there are, $5\times5=25$ ways to do this. The total number of ways to answer $8$ questions correctly in $10$ is $\binom{10}{8}=45$. Hence, the probability desired is $$P=\frac{25}{45}=\frac{5}{9}$$
• You have $5$ ways to pick a quesiton to make a mistake in the first $5$ ones, and $5$ ways to pick in the $5$ last ones. So, $5\times 5=25$ – Mateus Sampaio Dec 6 '13 at 14:04