A student is given a test consisting of $30$ multiple-choice questions. For each question there are $5$ possible answers of which one is correct. To pass, the student must answer at least $25$ questions correctly. On the first $20$ he knows the answers and for the remaining questions he chooses an answer haphazardly and independently. Calculate the probability that he will pass the test.
I solved this problem a few months ago but I forgot how I did it. Does anyone have a better suggestion than what I did?
$$\sum _{i=0}^5\frac{10!}{\left(5-i\right)!\left(10-\left(5-i\right)\right)!}\left(\frac{1}{5}\right)^{5+i}\left(\frac{4}{5}\right)^{5-i}=0.03279$$
HINT: Use the Binomial Distribution. There are $10$ remaining question and the student needs at least $5$ to be guessed correctly. Each question has a fixed probability of success $0.2$. Therefore, we need to find $\mathbb{P}(X \ge 5)$ where $X$~Binomial$(10,0.2)$. This is the same as $1- \mathbb{P}(X < 5)$. To find this probability, you can use the Cumulative Distribution Function for the Binomial Distribution to complete the problem (or you can calculate the sum of the Probability Mass Function over the values $0$ to $4$ (since the inequality is up to, but not including, the number value $X=5$) .