Question

In an examination, 20 questions of true-false type are asked. Suppose a student tosses a fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true’ ; if it falls tails, he answers ‘false’. Find the probability that he answers at least 12 questions correctly.

Solution

Here n = 20
p = P(answers true) = P(head) = $1 half$
$therefore space space space space straight q space equals space 1 minus straight p space equals space 1 minus 1 half space equals space 1 half$
P(answers at least 12 questions correctly)
= P(12) + P(13) + P(14) + P(15) + P (16) + P(17) + P(18) + P(19) + P(20)
$equals space straight C presuperscript 20 subscript 12 space open parentheses 1 half close parentheses to the power of 8 space open parentheses 1 half close parentheses to the power of 12 space plus space straight C presuperscript 20 subscript 13 space open parentheses 1 half close parentheses to the power of 7 space open parentheses 1 over 12 close parentheses to the power of 13 space plus space straight C presuperscript 20 subscript 14 space open parentheses 1 half close parentheses to the power of 6 space open parentheses 1 half close parentheses to the power of 14$ $plus space straight C presuperscript 20 subscript 15 space open parentheses 1 half close parentheses to the power of 5 space open parentheses 1 half close parentheses to the power of 15 space plus space straight C presuperscript 20 subscript 16 space open parentheses 1 half close parentheses to the power of 4 space open parentheses 1 half close parentheses to the power of 16 space plus space straight C presuperscript 20 subscript 17 space open parentheses 1 half close parentheses cubed space open parentheses 1 half close parentheses to the power of 17$
$plus space straight C presuperscript 20 subscript 18 space open parentheses 1 half close parentheses squared space open parentheses 1 half close parentheses to the power of 18 space plus space straight C presuperscript 20 subscript 19 space open parentheses 1 half close parentheses space open parentheses 1 half close parentheses to the power of 19 space plus space straight C presuperscript 20 subscript 20 space open parentheses 1 half close parentheses to the power of 20$
$equals space 1 over 2 to the power of 20 space open parentheses straight C presuperscript 20 subscript 12 plus straight C presuperscript 20 subscript 13 plus straight C presuperscript 20 subscript 14 plus straight C presuperscript 20 subscript 15 plus straight C presuperscript 20 subscript 16 plus straight C presuperscript 20 subscript 17 plus straight C presuperscript 20 subscript 18 space plus space straight C presuperscript 20 subscript 19 plus space straight C presuperscript 20 subscript 20 space close parentheses$

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