Question

Emmanuel is given a multiple-choice exam with ten questions and each question with
five 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

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

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

Given $n=10,p=0.2.$

$$P(X\geq 6)=P(X=6)+P(X=7)+$$

$$+P(X=8)+P(X=9)+P(X=10)=$$

$$=\binom{10}{6}(0.2)^6(1-0.2)^{10-6}+\binom{10}{7}(0.2)^7(1-0.2)^{10-7}+$$

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

$$+\binom{10}{10}(0.2)^{10}(1-0.2)^{10-10}=210(0.2)^6(0.8)^4+$$

$$+120(0.2)^7(0.8)^3+45(0.2)^8(0.8)^2+$$

$$+10(0.2)^9(0.8)+(0.2)^{10}+=$$

$$=0.005505024+0.000786432+0.000073728+$$

$$+0.000004096+0.0000001024\approx0.0064$$

The probability that he will get at least six questions correct is $0.0064.$