Probability of 25-30 Correct Answers on 200-Q Quiz

[etotheix]

Homework Statement

A multiple-choice quiz has 200 questions each with 4 possible answers of which 1 is correct. What is the probability that guesswork yields from 25 to 30 correct answers for 80 of the 200 questions. [HINT: approximation may be helpful here]

Homework Equations

Binomial distribution
(n choose x)p^x*(1-p)^(n-x)

The Attempt at a Solution

Since 80! is too big of a number and I can't calculate for instance (80 chose 25) I decided to divide everything by 5, I am not sure if this is what the hint is indicating.

Now I have 16 questions and I have to find the probability of answering correctly from 5 to 6.

Event A: answer 5 questions correctly
Event B: answer 6 questions correctly

I am looking for P(A OR B) = P(A) + P(B) - P(A AND B)

P(A)=(16 choose 5)(0.25)^5*(0.75)^11=0.18
P(B)=(16 choose 6)(0.25)^6*(0.75)^10=0.11
P(A AND B)=0

P(A OR B)=0.18 + 0.11 = 0.29 or 29%

Is this the correct way? I am mostly confused because of the "from 25 to 30" and I am not sure if the division by 5 is correct.

Thank you in advance.

 

[danago]

I think when it says use an approximation it means use the normal distribution as an approximation to the binomial distribution, not divide everything by 5.

 

[etotheix]

I think when it says use an approximation it means use the normal distribution as an approximation to the binomial distribution, not divide everything by 5.

Thank you, I believe you are absolutely right. I am fairly new to probability as you can see.