a test consists of 10 multiple choice questions, each having 4 possible answers, one of which is correct. assume that you guess all 10 answers. find the probability of getting less than 40%

The probability of getting any individual question right is 25%. So, the probability of getting specifically the first r questions right and missing the rest is 0.25^r * 0.75^(n-r). But there are more ways to get r questions right than just getting the first r. There are nCr ways of selecting the r questions to get right. So the total probability of getting r questions right is nCr*0.25^r * 0.75^(n-r).

n | probability

0 | 0.05631

1 | 0.18771

2 | 0.28157

3 | 0.25028

To get less than 40%, the student must get 0, 1, 2, or 3 questions right. The probability of that is calculated by looking at each number separately and adding. The result is 0.77588.