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(D) The total number of questions is $n = 8$. Since each question is true or false,the probability of answering correctly is $p = \frac{1}{2}$ and inc

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$\frac{219}{256}$

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(D) The total number of questions is $n = 8$. Since each question is true or false,the probability of answering correctly is $p = \frac{1}{2}$ and incorrectly is $q = \frac{1}{2}$.Let $X$ be the number of correct answers. $X$ follows a binomial distribution $B(8, \frac{1}{2})$.The student passes if $X \ge 6$.$P(	ext{Pass}) = P(X=6) + P(X=7) + P(X=8)$$P(	ext{Pass}) = \binom{8}{6}(\frac{1}{2})^8 + \binom{8}{7}(\frac{1}{2})^8 + \binom{8}{8}(\frac{1}{2})^8$$P(	ext{Pass}) = \frac{28 + 8 + 1}{256} = \frac{37}{256}$.The probability that the student fails is $P(	ext{Fail}) = 1 - P(	ext{Pass})$.$P(	ext{Fail}) = 1 - \frac{37}{256} = \frac{219}{256}$.

$A$ student writes an examination which contains $8$ true or false questions. If he answers $6$ or more questions correctly,he passes the examination. If the student answers all the questions,then the probability that he fails in the examination is

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