Answer to Question #185743 in Statistics and Probability for Mohammed Moro

Question

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 1

Let $X=$ the number of correct answers: $X\sim Bin(n, p).$

Probability of getting a correct answer is $p=\dfrac{1}{5}=0.2$

$n=15$

$$P(X\geq10)=P(X=10)+P(X=11)$$

$$+P(X=12)+P(X=13)+P(X=14)$$

$$+P(X=15)=\dbinom{15}{10}(0.2)^{10}(1-0.2)^{15-10}$$

$$+\dbinom{15}{11}(0.2)^{11}(1-0.2)^{15-11}$$

$$+\dbinom{15}{12}(0.2)^{12}(1-0.2)^{15-12}$$

$$+\dbinom{15}{13}(0.2)^{13}(1-0.2)^{15-13}$$

$$+\dbinom{15}{14}(0.2)^{14}(1-0.2)^{15-14}$$

$$+\dbinom{15}{15}(0.2)^{15}(1-0.2)^{15-15}$$

$$\approx0.0001132257$$