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Question
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`
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