Question

A multiple choice quiz has 400 questions, each question with five possible answers of which one is correct answer. What is the probability that a sheer guess yields exactly 80 correct answers using normal approximation.     

Answer 1

X ~ Bin(n,p)

$n=400 \\ p=\frac{1}{4}=0.2 \\ q=1-p=1-0.2=0.8 \\ mean = n \times p = 400 \times 0.2 =80 \\ \sigma = \sqrt{npq}=\sqrt{400 \times 0.2 \times 0.8}=8$

The probability that a sheer guess yields exactly 80 correct answers:

$P(X=80) = P(79.5≤X≤80.5) \\ = P(X≤80.5) -P(X<79.5) \\ =P(Z≤ \frac{80.5-80}{8}) -P(Z< \frac{79.5-80}{8}) \\ =P(Z≤0.0625) -P(Z< -0.0625) \\ = 0.5251 -0.4752 \\ = 0.0499$