Question

in a multiple choice test there are 10 questions for each question there is a choice of 4 answers only one of the which is correct. a student guesses at each of the answers, if he needs to obtain over half mark to pass, and the question carry equal weight, find the probability that he passes

Answer 1

Let $X=$ the number of correctly answered questions: $X\sim Bin(n, p)$

$$P(X=x)=\dbinom{n}{k}p^x(1-p)^{n-x}$$

Given

$$p=0.25,n=10$$

If a student guesses on each question, what is the probability that the student will pass the test?

$$P(X\geq6)=P(X=6)+P(X=7)+P(X=8)+$$

$$+P(X=9)+P(X=10)=$$

$$=\dbinom{10}{6}0.25^6(1-0.25)^{10-6}+\dbinom{10}{7}0.25^x(1-0.25)^{10-7}+$$

$$+\dbinom{10}{8}0.25^8(1-0.25)^{10-8}+\dbinom{10}{9}0.25^9(1-0.25)^{10-9}+$$

$$+\dbinom{10}{10}0.25^10(1-0.25)^{10-10}\approx$$

$$\approx0.0162220+0.0030899+0.0003862+$$

$$+0.0000286+0.0000010=0.019728\approx0.02$$

The probability that the student will pass the test is $0.02\ (2\%).$