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Question
In a 20-question true-false examination, suppose a student tosses a fair coin to determine his answer to each question. For every head, he answers 'true' and for every tail, he answers 'false'. Find the probability that he answers at least 12 questions correctly.
Solution
Let X denote the number of correct answers.
Then, X follows a binomial distribution with \[n = 20\]
\[\text{ Let p be the probability of a correct answer} . \]
\[ \Rightarrow p = \text{ getting a head and a right answer to be true or getting a tail and a right answer to be false } \]
\[ \Rightarrow p = \frac{1}{2}\]
\[ \therefore q = 1 - \frac{1}{2} = \frac{1}{2}\]
\[\text{ Hence, the distribution is given by } \]
\[P(X = r) = ^{20}{}{C}_r \left( \frac{1}{2} \right)^r \left( \frac{1}{2} \right)^{20 - r} , r = 0, 1, 2, 3 . . . . . . 20\]
\[ = \frac{^{20}{}{C}_r}{2^{20}}\]
\[\text{ Probability that the student answers at least 12 questions correctly } = P(X \geq 12) \]
\[ = P(X = 12) + P(X = 13) + . . . + P(X = 20)\]
\[ = \frac{^{20}{}{C}_{12} + ^{20}{}{C}_{13} + . . . +^{20}{}{C}_{20}}{2^{20}}\]
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