Question

A student is given a quiz with 10 true or false questions and he answers by sheer guessing. If X is the number of questions answered correctly write the p.m.f. of X. If the student passes the quiz by getting 7 or more correct answers what is the probability that the student passes the quiz?

Answer 1

Here n = 10

P(Success) = P(answer is correct) = `1/2`

∴ p = `1/2`

`\implies` q = `1 - 1/2 = 1/2`

Let X = Number of question answered correctly

Then `X ∼ B(n = 10, p = 1/2)`

∴ p.m.f. of X is given by

P(X = x) = p(x)

= `""^10C_x(1/2)^x(1/2)^(10 - x)`,

x = 0, 1, 2, ...,, 10

P(Student pass the quiz)

= P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10)

= `""^10C_7(1/2)^7(1/2)^3 + ""^10C_8(1/2)^8(1/2)^2 + ""^10C_9(1/2)^9(1/2) + ""^10C_10(1/2)^10`

= `(1/2)^10 [""^10C_7 + ""^10C_8 + ""^10C_9 + ""^10C_10]`

= `1/1024[(10 xx 9 xx 8)/3.2 + (10 xx 9)/2 + 10 + 1]`

= `1/2024[120 + 45 + 11]`

= `176/1024`

= 0.1718