Question

1. Consider still the 15-question multiple choice quiz that each question has 5 options (a,b,c,d,e) and you randomly guess every question.
   1. How many questions does one need to answer correctly in order score exactly a 60%
2. 2. If a grade of 60% or lower is considered failing, then what is the probability of you failing?
3. 3. If you need a grade of 80% or higher on this quiz to maintain a passing grade, what is the probability of you maintaining that passing grade?

Answer 1

P(correct guessing) = 1/5 = 0.2 [as there is only one correct answer out of 5 options]

so, p=0.2

n=15

it is a binomial probability distribution,  
  
and probability is given by  

P(X=x) = C(n,x)*px*(1-p)(n-x)

a)

to score exactly 60%, one to need to answer 15*0.60 = 9 answers correctly

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b)

grade of 60% or lower means, one has answered 9 or fewer correct answer,so

P( X ≤ 9) = ΣC(n,x)*p^x*(1-p)^(n-x) where x goes from 0 to 9

P(x≤9) =0.9999
So, probability of you failing = 0.9999

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c)

80% score means = 0.80*15 = 12 correct answer

so,

P(X≥12) =1-P(x<12) = 1 - ΣC(n,x)*p^x*(1-p)^(n-x) where x goes from 0 to 11

P(x≥12) =0.00000101

probability of you maintaining that passing grade=0.00000101