Final answer:
The probability of guessing correctly on a multiple-choice question with five options can be calculated using the binomial probability formula. For exactly four correct guesses out of 16, specific values are substituted into this formula. The standard deviation can be found using the formula for the binomial distribution.
Step-by-step explanation:
To calculate the probability of a student guessing correctly on a multiple choice question with five options, we use the binomial probability formula, which is applicable since each question is independent, and the probability of guessing right or wrong is the same for each question. The formula is P(x) = (n choose x) * p^x * (1-p)^(n-x), where n is the number of trials (questions), x is the number of successes (correct guesses), and p is the probability of success on a single trial.
For part (a), if we want to calculate the probability of exactly four correct guesses out of 16 questions, we substitute 16 for n, 4 for x, and 1/5 for p, since there is a 1 in 5 chance of guessing a question correctly. To calculate part (b), which asks between three and eight answers correct inclusive, we would sum the probabilities for each value of x from 3 to 8.
Part (d) asks for the standard deviation, which measures the dispersion of the probability distribution. The formula for the standard deviation of a binomial distribution is σ = √(np(1-p)), where n is the number of trials and p is the probability of success. Substituting 16 for n and 1/5 for p, we can calculate the standard deviation.
Ignoring the irrelevant portions of the question, we focus on the multiple-choice exam probability and standard deviation problems.