Question

Each of the total five questions in a multiple choice examination has four choices, only one of which is correct. A student is attempting to guess the answer. The random variable x is the number of questions answered correctly. What is the probability that the student will give atleast one correct answer?

Answer 1

Let X : number of question answered correctly out of 5.
p = P (correct answer)
p = 1/4
q = 1 - p = 3/4 and n = 5

X ~ B( n= 5,p =1/4)

The p.m.f of X is given as

`p(X = x) = p(x) =""^5C_x(1/4)^x(3/4)^5-x`

where x = 0, 1, 2, 3, 4, 5
Probability a student will get at least one correct answer

= P(X ≥ 1) = 1 - P(X < 1) 
= 1 - P(X = 0)

=

`1-""^5C_0(1/4)^0(3/4)^5`

=

`1-243/1024`

=781/1024=0.7627