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  • The probability that a student will pass an examination of 100 questions is 90 %. If the student answers the questions at random, what is the probability that he will answer at least 80 questions c

The probability that a student will pass an examination of 100 questions is 90 %. If the student answers the questions at random, what is the probability that he will answer at least 80 questions correctly?

Option: 1

0.24


Option: 2

0.31


Option: 3

0.18


Option: 4

0.08


Answers (1)

best_answer

The probability that a student will answer a question correctly is 0.9 and the probability that he will answer a question incorrectly is 0.1

The probability that he will answer at least 80 questions correctly is equal to the sum of the probabilities that he will answer:
Exactly 80 questions correctly: \mathrm{(0.9)^{\wedge} 80 * 0.1^{\wedge} 20}
Exactly 81 questions correctly:\mathrm{ (0.9)^{\wedge} 81^{\star} 0.1^{\wedge 19}}
Exactly 82 questions correctly: \mathrm{ (0.9) \wedge 82 * 0.1 \wedge 18}
Exactly 100 questions correctly: \mathrm{(0.9)^{\wedge} 100}

The sum of these probabilities can be calculated using the Binomial Theorem.

\mathrm{n \operatorname{Cr}(p)^{\wedge} r(q)^{\wedge} n-r}
Where\mathrm{ n=100, r=80, p=0.9} and  \mathrm{q=0.1}
The sum of the probabilities is:

\mathrm{\begin{array}{r} 100 \mathrm{C} 80(0.9)^{\wedge} 80(0.1)^{\wedge} 20+100 \mathrm{C} 81(0.9) \wedge 81 \\ (0.1)^{\wedge} 19+\ldots+100 \mathrm{C} 100(0.9) \wedge 100=0.18 \end{array}}

Thus, the probability that he will answer at least 80 questions correctly is 0.18 .

 

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Rishabh

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