An exam has 20 questions, which a student answers either correctly or incorrectl

Question

An exam has 20 questions, which a student answers either correctly or incorrectly. A certain student who (very wisely) does study for this exam has an 85% chance on each question of getting a correct answer, independently of all the other questions. What is the student's probability of getting at most one incorrect answer? Suppose that the student runs into the instructor in the hallway the day after the exam, and asks. "Do you know my grade yet?" Not wanting to give a complete answer, but wanting to say something reassuring, the instructor replies, "I can tell you that you got at least 80% correct." In light of this new information, what is the student's probability of getting at most one incorrect answer?

Explanation / Answer

p(correct) = .85
p(incorrect) = .15
no. of questions = 20

1a. P( atmost 1 incorrect answer)
= 20C20(.15^0)(.85^20)+20C19(.15^19)(.85^1)
= 0.1755

1b. P(atleast 80% correct)=?
P(X>=16)=P(X=16,17,18,19,20)
= 0.8298

P(getting atmost 1 incorrect|given 80% atleast correct)
= 0.1755/.8298
=.2115

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