Question

Richard has just been given a 10-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 ten 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?

They should not be equal, but are equal.They should be equal, but may not be due to table error.    They should be equal, but differ substantially.They should be equal, but may differ slightly due to rounding error.

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

Answer 1

Answer:

a)

Given,

n = 10

p = 1/4

q = 1 - 1/4

= 3/4

To determine the probability that he will answer all questions correctly

let us utilize the binomial distribution

P(X = x) = nCr*p^r*q^(n-r)

P(X = 10) = 10C10*(1/4)^10*(3/4)^(10-10)

= (1/4)^10

= 0.25^10

b)

To determine the probability that he will answer all questions incorrectly

P(X = x) = 1 - P(x = 10)

= 1 - 0.25^10

= 1 - 0

= 1

c)

To determine the probability that he will answer at least one of the questions correctly

P(x >= 1) = 1 - P(x < 1)

= 1 - P(x = 0)

= 1 - 10C0*(1/4)^0*(3/4)^10

= 1 - (3/4)^10

= 1 - 0.75^10

= 1 - 0.0563

= 0.9437

Mostly equal to one but due to rounding error they may differ

d)

To determine the probability that Richard will answer at least half the questions correctly

P(x >= 5) = 1 - P(x < 5)

= 1 - [P(0) + P(1) + P(2) + P(3) + P(4)]

= 1 - [10C0*0.25^0*0.75^10 + 10C1*0.25^1*0.75^9 + 10C2*0.25^2*0.75^8 + 10C3*0.25^3*0.75^7 + 10C4*0.25^4*0.75^6]

= 1 - [0.0563 + 0.0563 + 0.2815 + 0.2502 + 0.1459]

= 1 - 0.7902

= 0.2098