A multiple choice test contains 12 questions each offering 4 answers, only one of which is correct each time. A student knows the correct asnwer to 7 of the questions but randomly guesses the anwers to the remaining five. What is the probability the student will get at least 10 of the twelve correct.
There are 4 answers for each multiple choice question and each of them has only one correct answer. If any answer is chosen at random the probability of it being correct is 1/4. The probability that it is incorrect is 3/4.
In the 12 question test, the student knows the correct answer to 7 of the questions. For the other 5 questions the answers are picked at random. To get at least 10 answers right, the student should get either 3 or 4 or 5 answers correct for the 5 questions she does not know the answer to.
The probability of this is `(1/4)^3*(3/4)^2` + `(1/4)^4*(3/4)` + `(1/4)^5` = `13/1024`
The required probability is `13/1024`