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In a multiple-choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing? - Mathematics

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Sum
In a multiple-choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
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Solution

The repeated guessing of correct answers from multiple-choice questions is Bernoulli trials. Let X represent the number of correct answers by guessing in the set of 5 multiple-choice questions.

Probability of getting a correct answer is, p `= 1/3`

` ∴ "q" = 1 - "p" = 1 - 1/3 = 2/3`

Clearly, X has a binomial distribution with n = 5 and p `= 1/3`

∴ p (X = x) = `""^"n""C"_"x" "q"^("n"-"x")"p"^"x"`

` = ""^5"C"_"x" (2/3)^(5-"x").(1/3)^"x"`

P (guessing more than 4 correct answers) = P(X ≥ 4)

 = P (X = 4)+ (X = 5)

` = ""^5"C"_4(2/3).(1/3)^4 + ""^5"C"_5(1/3)^5`

` = 5. 2/3 . 1/81 + 1 . 1/243`

` = 10/243 + 1/243`

` = 11/243`

Concept: Bernoulli Trials and Binomial Distribution
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