Question

# In a 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'. The probability that he answers at least 12 questions correctly is:

A
(12)20(22020C102)
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B
(12)20(22020C10220C92)
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C
(12)20(22020C1020C92)
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D
(12)20(220220C92)
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Solution

## The correct option is B (12)20(220−20C10−2⋅20C92)Let us assume that the number of correctly answered questions out of twenty questions be X. Since,'head' on the coin shows the true answer and the 'tail' on the coin shows the false answers. Thus, the repeated tosses or the correctly answered questions are Bernoulli trials. Thus, p=1/2 and q=1−p=1−1/2=1/2 Here, it can be clearly observed that X has binomial distribution, where n=20 and p=1/2 Thus, P(X=x)=nCxqn−xpx, where x=0,1,2,...n=20Cx(12)20−x(12)x=20Cx(12)20 Probability of at least 12 questions answered correctly =P(X≥12)=P(X=12)+P(X=13)+⋯+P(X=20)) =20C12(12)20+20C13(12)20+⋯+20C20(12)20=(12)20(20C12+20C13+...+20C20) =(12)20(220−20C10−2⋅20C92) {∵20C0+20C1+⋯+20C20=220⇒20C10+2(20C11+20C12+⋯+20C20)=220⇒20C12+⋯+20C20=220−20C10−2⋅20C92}

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