Question

In a multiple choice test there are 10 questions and for each question there is a choice of 4
answers, only one of which is correct. If a student guesses at each of the answers, find the
probability that he gets
(a) None correct, [3]
(b) More than 7 correct, [4]
If he needs to obtain over half marks to pass, and the questions carry equal weight, find the
probability that he passes. [4]

Answer 1

a) Probability that the student answers a question incorrectly is $\frac{3}{4}$. Then probability that the student answers 10 questions incorrectly is $(\frac{3}{4})^{10}\approx 0.056$ (events "the student answers the 1_st question, the 2_nd question,$\ldots$, the 10_th question incorrectly" are independent collectively).

b) We should find probability that the student answers 8, 9 or 10 questions correctly. So probability is $\sum_{k=8}^{10}C_{10}^k(\frac{1}{4})^k(\frac{3}{4})^{10-k}\approx 0.000416$ (Bernoulli's formula).

c) We should find probability the the student answers 6, 7, 8, 9 or 10 questions correctly.

So probability is $\sum_{k=6}^{10}C_{10}^k(\frac{1}{4})^k(\frac{3}{4})^{10-k}\approx 0.0197$ (Bernoulli's formula).