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.)