a test consists of 10 multiple choice questions each with five possible answers, one of which is correct. to pass the test a student must get 60% or better on the test. if a student randomly guesses, what is the probability that the student will pass the test?

Question

kirbykirby · Answer

This is a Binomial distribution type of problem. You consider that the 10 questions are 10 trials. Since the student is randomly guessing, you assume that each guess is independent of each other. And the success probability is the probability of obtaining one question correct , i.e. $p=1/5$

So, if $X$ is the number of questions a student guesses correctly, then $X\sim Binomial(10, \frac{1}{5})$
To get 60% or better on the test means the person must get 6 questions or greater correctly on the exam. Thus, you need to find $P(X\ge 6)$

$$ P(X\ge 6)=P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)\
= { 10 \choose 6}\left(\frac{1}{5} \right)^6\left(1-\frac{1}{6} \right)^4+\ldots +{ 10 \choose 10}\left(\frac{1}{5} \right)^{10}\left(1-\frac{1}{6} \right)^0$$