A quiz contains 8 multiple choice questions, each with 4 choices. a. If you guess completely at random on one question, what is your probability p of getting the question right? b. For the quiz, you r

a. The probability of guessing the correct answer to one multiple choice question with 4 options is 1/4, since there is only one correct answer out of the four choices. b. This is a binomial experiment with n=8, the number of trials (questions) and p=1/4, the probability of success (getting one question right). c. The four conditions satisfied for a binomial experiment are: 1. Each trial (question) has only two possible outcomes: success (getting the question right) or failure (getting the question wrong). 2. The trials are independent of each other, meaning the outcome of one question does not affect the outcome of another question. 3. The probability of success (p) is constant for all trials. 4. The number of trials (n) is fixed and known in advance. d. P(X = 6) can be calculated by the binomial probability formula: P(X=k) = (n choose k) * p^k * (1-p)^(n-k). Plugging in the values, we get P(X=6) = (8 choose 6) * (1/4)^6 * (3/4)^2 = 0.0107 (by hand). e. The probability of getting at least 7 questions correct can be calculated by adding the probabilities of getting 7, 8 correct. Using the binomial probability formula for each value of k and summing the probabilities, we get P(X ? 7) = P(X=7) + P(X=8) = (8 choose 7) * (1/4)^7 * (3/4)^1 + (8 choose 8) * (1/4)^8 * (3/4)^0 = 0.0006 (by hand). Using the binomial cumulative distribution function in a table, we get P(X ? 7) = 0.0004. f. P(X ? 4) can be calculated using a table or Excel by summing the probabilities of getting 0, 1, 2, 3, or 4 questions correct. Using a table, we find P(X ? 4) = 0.433. Using Excel, we can use the formula =BINOM.DIST(4,8,0.25,TRUE) to get the same answer. g. The probability of getting fewer than 3 questions correct can be calculated by summing the probabilities of getting 0, 1, or 2 questions correct. Using the binomial probability formula for each value of k, we get P(X < 3) = P(X=0) + P(X=1) + P(X=2) = (8 choose 0) * (1/4)^0 * (3/4)^8 + (8 choose 1) * (1/4)^1 * (3/4)^7 + (8 choose 2) * (1/4)^2 * (3/4)^6 = 0.557 (by hand). h. The mean of a binomial distribution is n*p, which in this case is 8*(1/4) = 2. The variance of a binomial distribution is n*p*(1-p), which in this case is 8*(1/4)*(3/4) = 1.5. The standard deviation is the square root of the variance, which is \approx imately 1.22.