Answer: 0.00011323
Step-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
