How to calculate probability that an examinee gets X questions correct by pure guessing in a multiple-choice exam?

Question

What is the chance that an examinee gets at least 7 questions correct by pure guessing in a multiple-choice exam with 14 questions and 4 possible answers per question?

Answer 1

The probability of a correct guess is in any individual question is ¼ (0.25) so ¾ (0.75) is the probability of a wrong guess. This is a binomial situation.

If p=0.25 and q=0.75 then (p+q)¹⁴=1 represents all the outcomes. We can use the binomial distribution or use the normal distribution approximation. Remember that the normal distribution is continuous, whereas the binomial distribution is discrete.

The binomial expansion is:

The p-q terms are:

p¹⁴+14p¹³q+91p¹²q²+364p¹¹q³+1001p¹⁰q⁴+2002p⁹q⁵+3003p⁸q⁶+3432p⁷q⁷ corres-ponding respectively to the probability of exactly the number of correct answers out of 14 questions:

14 right, 13 right, 12 right, 11 right, 10 right, 9 right, 8 right, 7 right.

This evaluates to 0.03827 approx. So the probability of getting at least 7 answers correct out of 14 is about 3.83%. (The mean result is 3.5 out of 14.)

Comments

Hello Rod, thank you for your help as always.