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 com falls heads, he answers "true1; if it falls tails, he answers "false1. Find the probability th

Options

A. `((1)/(2))^(20)(""^(20)C_(12)+""^(20)C_(13)+........+""^(20)C_(20))`
B. `((1)/(2))^(10)(""^(20)C_(11)+""^(20)C_(12)+........+""^(20)C_(20))`
C. `((1)/(2))^(20)(""^(20)C_(11)+""^(20)C_(12)+........+""^(20)C_(20))`
D. None of the above

Answer

A

Explanation

Let X denotes the number of correct answers given by the student . The repeated tosses of a coin are Bernoulli trails.Since, head on a coin represents the true answer and tail represents the false answer, the correctly answered of the question are Bernoulli trials.
`therefore ` p=P(a success)=P(coin show up a head)=`1/2`
`therefore q=1-p=1-1/2=1/2`
`therefore ` X has a binomial distribution with
`n=20, p=1/2 and q=1/2`
`therefore P(X=r)=""^(20)C_(r).((1)/(2))^(r) ((1)/(2))^(20-r)`
Hence , P (atleast 12 questions are answered as true )
`=P(X ge 12) = P(x=12 ) + P(x=13) + P(x+14)+P(x=15)+P(x=16)+P(x=17)+P(x=18)+P(x=19)+P(x=20)`
`""^(20)C_(12)p^(12)q^(8)+""^(20)C_(13)p^(13)q^(7)+""^(20)C_(14)p^(14)q^(6)+""^(20)C_(15)p^(15)q^(5)+""^(20)C_(16)p^(16)q^(4)+""^(20)C_(17)p^(17)q^(3)+""^(20)C_(18)p^(18)q^(2)+""^(20)C_(19)p^(19)q^(1)+""^(20)C_(20)P^(20)`
`=(""^(20)C_(12)+""^(20)C_(13)+""^(20)C_(14)+""^(20)C_(15)+""^(20)C_(16)+""^(20)C_(17)+""^(20)C_(18)+""^(20)C_(19)+""^(20)C_(20)).(1)/(2^(20))`
`(1/2)^(20)[""^(20)C_(12)+""^(20)C_(13)+.....+""^(20)C_(20)]`