An exam has 20 multiple choice questions, each question has 2 possible answers,

Question

An exam has 20 multiple choice questions, each question has 2 possible answers, and only one of them is correct. A student will answer all questions completely at random, independently of one another. What is the probability that:

(a) the first 10 answers are correct and the next 10 answers wrong?

(b) there at at least 4 and at most 16 correct answers?

(c) Suppose the student gets 3 points for each correct answer and loses 1 point for each wrong answer. Call X the final score of the student. What are the values that this random variable takes? Is X discrete or continuous?

Explanation / Answer

Here, p = 1/2

a)

p^10 * ( 1 - p)^10

= (1/2)^10 * ( 1 - 1/2)^10

= 0.0019

b)

Required probability = 1 - (20C0 * p^0 * ( 1-p)^20 + 20C1 * p^1 * ( 1-p)^19 + 20C2 * p^2 * ( 1-p)^18 +20C3 * p^3 * ( 1-p)^17) - (20C17 * p^17 * ( 1-p)^3 + 20C18 * p^18 * ( 1-p)^2 + 20C19 * p^19 * ( 1-p)^1 + 20C20 * p^20 * ( 1-p)^0)

= 1- 0.00128 - 0. 00128

= 0.9974

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