there is a 12 question multiple choice test. Each question has four possible choices, one of which is the correct answer. An unprepared student randomly guesses at each question.

1. what is the exact number of possible correct answers?
2. What is the standard deviation for the number of possible correct answers?
3. Probability of getting exactly 6 correct answers?
4. Probabilty of getting at least 6 correct answers?

1. Come on, the first part is simple. There are 12 questions... 4 options, and only one good answer for each question.

2. Taking the problem as a binomial distribution, the standard deviation is given by \sigma = \sqrt{np}

from X ~ B(n, p)
where X is the event denoting a correct answer,
n is the number of 'successes' and p is the probability of each 'success'

3. P(X = 6) = \(12\\ 6\) (1-p)^6(p)^6

4. At least six means that there are 6, 7, 8, 9, 10, 11 or 12 correct answers.

P(X \geq 6) = P(X = 6) + P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) + P(X = 11) + P(X = 12)