Exercise 3:

A true-false examination has 48 questions. June has probability 3/4 of answering a question correctly. April just guesses on each question. A passing score is 30 or more correct answers. Compare the probability that June passes the exam with the probability that April passes it.

Answer :

Given , n =48

We need to find P(S48 >=30)

This can be modelled using the binomial distribution and approximated using the Central Limit Theorem. \[ P( i≤ Sn ≤j ) ≈ NA( (i- 1/2 -np)/sqrt(npq) , (j +1/2 -np)/sqrt(npq) ) \]

For June,
p =3/4 and q =1/4

P(S48 >=30) = (30 -36 - .5)/3 = -2.166667

# Probability that June pass the exam
1- pnorm(-2.166667)
## [1] 0.9848699

The probability that June pass the exam is 0.9848699
For April ,  

p=1/2 and q =1/2
P(S48 >=30) = (30 -24 - .5)/sqrt(48(1/2)(1/2)) = 1.587713

 1-pnorm((30-24-0.5)/(2*sqrt(3)))
## [1] 0.0561756

The probability that April pass the exam is 0.056175598845231933