Question

Emmanuel is given a multiple- choice exam with 10 questions & each question with 5 possible answers .He decided to guess randomly for each question.What is the probability that he will get at least six questions correct?

Answer 1

A question has 5 possible choices out of which only one choice is correct. So, probability of choosing the correct answer , p=1/5=0.2

Since there are total 10 questions, n=10

Let X be the number of correct answers.

Then X follows Binomial distribution with n=10 and p=0.2

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

Probability that at least six questions are correct will be

P(X $\geq6)$=P(X=6)+P(X=7)+P(X=8)+P(X=9)+P(X=10)

=$\binom{10}{6}(0.2)^6(1-0.2)^4 +$

$\binom{10}{7}(0.2)^7(1-0.2)^3 +$

$\binom{10}{8}(0.2)^8(1-0.2)^2+$ $\binom{10}{9}(0.2)^9(1-0.2)^1+$ $\binom{10}{10}(0.2)^{10}(1-0.2)^0$

= 210.(0.00026) + 120.(0.0000065)+ 45(.0000016) + 10.(.00000041) + 1.(0.2)¹⁰

=0.006369