A final exam of the course Probability 101 consists of 10 multiple-choice questions. Each question has 4 possible answers and only one of them is a correct answer. To pass the course, 8 or more correct answers are necessary. Assume that a student has not studied probability at all and has no idea how to solve the questions. So the student decided to answer each questions randomly. Thus, for each of 10 questions, the student choose one of the 4 answers randomly and each choice is independent each other.
(1) What is the probability that the student answered correctly only one question among the 10 questions?
(2) Determine the probability that the student passes the course.
(3) What is the expected value of the number of questions the student answered correctly?
(4) Find the variance and standard deviation of the number of questions the student answered correctly.
A random variable $X$ is said to be a Bernoulli random variable if its probability mass function is given by
P(X=0) &= 1-p\\
P(X=1) & = p
for some real number $0 \leq p \leq 1$.
(1) Find the expectation of the Bernoulli random variable $X$ with probability $p$.