A single choice examination has 10 questions, each with five possible answers, only one is correct. Suppose a student randomly guesses at each question.
a.) What is the probability the student answers no questions correct?
b.) What is the probability that the student passes the test (60% or better)
c.) What is the probability that the student's first correct answer is on the seventh question?
My thoughts
a.) (4/5)^10 since the probability of getting each question wrong is the same
b.)
$$P(Y\ge6)=P(Y=6)+ P(Y=4)$$
$$ \left(^{10}_6\right)(.2)^{6}(.8)^{4}+\left(^{10}_{10}\right)(.2)^{10}(.8)^{10-10}$$
$$P(Y\ge6)=.0055$$
c.)(4/5)^6 x (1/5)
Best Answer
A straightforward (if long-ish) way to answer the second question is to consider the probability that the person gets exactly $6,7,8,9,$ and $10$ questions correct.
There are ${10 \choose n}$ ways that the student can get $n$ questions correct. The probability that this happens is $(1/5)^n(4/5)^{10-n}$. Then your answer is
$$P_{pass} = \sum_{n=6}^{10}{10 \choose n}\left(\frac15\right)^n\left(\frac45\right)^{10-n}$$