Richard has just been given a 6-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 six questions, find the indicated probabilities. (Round your answers to three decimal places.) What is the probability that Richard will answer at least half the questions correctly?

0.169

Step-by-step explanation:

This is a question on binomial distribution where we only have two outcomes: success or failure

Binomial distribution formula:

P(X= x) = n!/[x!(n-x)!] p^x × q^(n-x)

n is the number of questions in the quiz n = 6

Probability of guessing a correct number = probability of success = p

We told each question has 4 options in which one is correct.

The probability of guessing a correct answer = p= 1/4

p = 0.25

q = 1-p = 1-0.25

q = 0.75

Probability that Richard will answer at least half the questions correctly = P(X≥3)

P(X≥3) = P(X=3) +P(X=4) +P(X=5) +P(X=6)

See attachment for details

P(X≥3) = 0.01318 + 0.0330 + 0.0044 + 0.0002

P(X≥3) = 0.1694

The probability that Richard will answer at least half the questions correctly = 0.169