## Passing an Exam on Pure Luck

Some students do not study at all and imagine that craft and guile or pure good luck will get them through an exam. What is the probability of passing an exam on good luck alone?
A multiple choice exam has 40 questions, all multiple choice. Each question contains 4 options, of which one gets full marks, and the other three get no marks. The pass mark is 24 out of 40,
If the correct answers are randomly distributed amongst the four options, and the student guesses randomly, they will guess correctly one quarter of the time. We can model the student's score by a binomial distribution
$B(40,0.25)$
.,
We calculate
$P(X \ge 24)=1-P(X \le 23) =1-0.99999717=0.00000283$
.
Highly unlikely.