An examination consists of 10 multiple choice questions, in each of which a candidate has to deduce which one of five suggested answers is correct. - Mathematics and Statistics

Sum

An examination consists of 10 multiple choice questions, in each of which a candidate has to deduce which one of five suggested answers is correct. A completely unprepared student guesses each answer completely randomly. What is the probability that this student gets 8 or more questions correct? Draw the appropriate morals.

Solution

Let X = number of correct answers.

p = probability that student gets a correct answer

∴ p = 1/5

∴ q = 1 - p = 1 - 1/5 = 4/5

Given: n = 10 (number of total questions)

∴ X ~ B (10, 1/5)

The p.m.f. of X is given by

P[X = x] = "^nC_x  p^x  q^(n - x)

i.e. p(x) = "^10C_x  (1/5)^x  (4/5)^(10 - x), x = 0, 1, 2,...,10

P(student gets 8 or more questions correct)

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

= ""^10C_8 (1/5)^8 (4/5)^2 + ""^10C_9 (1/5)^9 (4/5)^1 + "^10C_10 (1/5)^10 (4/5)^0

= (10 xx 9 xx 8!)/(8! xx 2 xx 1) xx (1/5)^8 xx (4/5)^2 + 10(1/5)^9 (4/5)^1 + 1 xx (1/5)^10

= 45 xx (1/5)^8 xx (4/5)^2 + 10 xx (1/5)^9 xx (4/5) + (1/5)^10

= (1/5)^8 [45 xx (4/5)^2 + 10 xx (1/5) xx (4/5) + (1/5)^2]

= [45 xx 16/25 + 10/5 xx 4/5 + 1/25](1/5)^8

= [720/25 + 40/25 + 1/25](1/5^8)

= (761/25) xx (1/5^8) = 30.44/5^8

Hence, the probability that student gets 8 or more questions correct = 30.44/5^8

Concept: Binomial Distribution
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APPEARS IN

Balbharati Mathematics and Statistics 2 (Arts and Science) 12th Standard HSC Maharashtra State Board
Chapter 8 Binomial Distribution
Miscellaneous exercise 8 | Q 10 | Page 254