Richard has just been given a 4-question multiple-choice quiz in his history cla

Question

Richard has just been given a 4-question multiple-choice quiz in his history class. Each question has five 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?
         
    
    (b) What is the probability that he will answer all questions incorrectly?
         
    
    (c) What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table.
         
    
    Then use the fact that P(r ? 1) = 1 ? P(r = 0).
         
    
    Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference?

    (d) What is the probability that Richard will answer at least half the questions correctly?
    

Richard has just been given a 4-question multiple-choice quiz in his history class. Each question has five 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.) What is the probability that he will answer all questions correctly? What is the probability that he will answer all questions incorrectly? What is the probability that he will answer at least one of the questions correctly? Compute this probability two ways. First, use the rule for mutually exclusive events and the probabilities shown in the binomial probability distribution table. Then use the fact that P(r? 1) = 1 ? P(r = 0). Compare the two results. Should they be equal? Are they equal? If not, how do you account for the difference? They should be equal, but may differ slightly due to rounding error. They should not be equal, but are equal. They should be equal, but differ substantially. They should be equal, but may not be due to table error. What is the probability that Richard will answer at least half the questions correctly?

Explanation / Answer

each question has five answers

so probability of guessing a question right=1/5

so probability of guessing a question wrong=4/5

NOTE : P(0)+P(1)+P(2)+P(3)+P(4)=1

a)there are 4 questions

so prob(all right)=1/5*1/5*1/5*1/5=1/625=0.0016

b)so prob(all wrong)=4/5*4/5*4/5*4/5=256/625=0.4096

c)method 1:

probability that he will answer at least one of the questions correctly=P(1)+P(2)+P(3)+P(4)= 4Cx (1/5)^x (4/5)^(4-x)..(x=1 to 4)

=4C1 (1/5)^1(4/5)^3 + 4C2 (1/5)^2(4/5)^2 + 4C3(1/5)^3(4/5)^1 + 4C4 (1/5)^4(4/5)^0=369/625=0.5904

method 2:

probability that he will answer at least one of the questions correctly=1-prob(he will answer 0 of the questions correctly)=1-P(0)=1-4/5*4/5*4/5*4/5=1-0.4096=369/625=0.5904

yes they should be equal.

d)probability that Richard will answer at least half the questions correctly=p(2)+p(3)+p(4)=1-p(0)-p(1)

p(o)=0.4096 ...from(b)

p(1)=4C1 (1/5)^1(4/5)^3=0.4096

so probab=1-p(0)-p(1)=113/625=0.1808

Related Questions

Navigate

• Browse (All)
• Browse R
• Subjects

---

---

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.