Question

There are 20 questions in a multiple choice driver competition. In each question there are 4 answers of which only one is correct. The score on each question is 0.5 points for each correct answer and 0 for each mistake. A candidate must pass at least 6 credits to pass the exam. If we assume that a candidate is lucky enough to answer all the questions in this test, calculate:
A. The probability that the candidate will "pass".
B. The probability that the candidate will get 10.
C. The average score we expect the candidate to get and a standard deviation.

Answer 1

Solution:
Given in the question
number of multiple-choice questions (n)= 20
Each question have 4 answers
So P(Correct) = 1/4 = 0.25
Candidate must pass at least 6 credits to pass the exams that mean candidate must give 12 correct answers to pass
Solution(a)
We need to calculate that the candidate will pass that means the candidate must give at least 12 correct answers, So we will use binomial distribution probability
P(X=n |N,p) = NCn*(p^n)*((1-p)^(N-n))
P(X>=12 |20,0.25) =  P(X=12) + P(X=13) + P(X=14) + P(X=15) + P(X=16) + P(X=17) + P(X=18) + P(X=19) + P(X=20) = 20C12*(0.25^12)*(1-0.25)^8 + 20C13*(0.25^13)*(1-0.25)^7 + 20C14*(0.25^14)*(1-0.25)^6 + 20C15*(0.25^15)*(1-0.25)^5 + 20C16*(0.25^16)*(1-0.25)^4 + 20C17*(0.25^17)*(1-0.25)^3 + 20C18*(0.25^18)*(1-0.25)^2 + 20C19*(0.25^19)*(1-0.25)^1 + 20C20*(0.25^20)*(1-0.25)^0 =  0.0009
So there is 0.9% probability that candidate pass the exam.
Solution(b)
We need to calculate the probability that the candidate will get 10 that means candidate must give all correct answer
P(X =20) = 20C20*(0.25)^20*(0.75)^0 = 0.0000001

Solution(c)
Expected score = n*p = 20*0.25 = 5
Standard deviation = sqrt(n*p*q) = sqrt(20*0.25*(1-0.25)) = sqrt(20*0.25*0.75) = 1.94