A multiple-choice examination has 15 questions, each with five possible answers, only one of which is correct. Suppose that one of the students who takes the examination answers each of the questions with an independent random guess. What is the probability that he answers at least ten questions correctly?

Answer: 0.00011323Step-by-step explanation:Given : A multiple-choice examination has 15 questions, each with five possible answers, only one of which is correct.i.e. Probability of getting a correct answer = [tex]p=\dfrac{1}{5}=0.2[/tex]Using Binomial probability formula , [tex]P(x)=^nC_xp^x(1-p)^{n-x}[/tex]The probability that he answers at least ten questions correctly will be :-[tex]P(x\geq10)=P(10)+P(11)+P(12)+P(13)+P(14)+P(15)\\\\=^{15}C_{10}(0.2)^{10}(0.8)^{5}+^{15}C_{11}(0.2)^{11}(0.8)^{4}+^{15}C_{12}(0.2)^{12}(0.8)^{3}+^{15}C_{13}(0.2)^{13}(0.8)^{2}+^{15}C_{14}(0.2)^{14}(0.8)^{1}+^{15}C_{15}(0.2)^{15}(0.8)^{0}\\\\=\dfrac{15!}{10!(15-10)!}(0.2)^{10}(0.8)^{5}+\dfrac{15!}{11!(15-11)!}(0.2)^{11}(0.8)^{4}+\dfrac{15!}{12!(15-12)!}(0.2)^{12}(0.8)^{3}+\dfrac{15!}{13!(15-13)!}(0.2)^{13}(0.8)^{2}+(15)(0.2)^{14}(0.8)^{1}+(1)(0.2)^{15}\\\\=0.000113225662464\approx0.00011323[/tex]Hence, the probability that he answers at least ten questions correctly = 0.00011323