Richard has just been given a 4-question multiple-choice quiz in his history class. each question has four answers, of which only one is correct. since richard has not attended class recently, he doesn't know any of the answers. assuming that richard guesses on all four questions, find the indicated probabilities. (round your answers to three decimal places.) (a) what is the probability that he will answer all questions correctly

To solve the following problem, we must use the binomial probability equation. This is expressed mathematically as:

P (r) = [n! / (n-r) ! r!] * p^r * q^ (n-r)

where,

n = total number of questions = 4

r = number of correct questions = 4

p = probability of success = 25% = 0.25

q = probability of failure = 0.75

Therefore substituting the values into the given equation will give us:

P (r = 4) = [4! / 0! 4!] * 0.25^4 * 0.75^0

P (r = 4) = 3.906 * 10^-3 = 0.391%

Answer: Richard only has 0.391% to answer all questions correctly by guessing.