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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

Let $$X$$ represent the number of correctly answered questions out of 20 questions.

The repeated tosses of a coin are Bernoulli trails. Since "head" on a coin represent the true answer and "tail" represents the false answer, the correctly answered questions are Bernoulli trials.

$$\therefore p=\displaystyle\frac { 1 }{ 2 } $$

$$\therefore q=1-p=1-\displaystyle\frac { 1 }{ 2 } =\displaystyle\frac { 1 }{ 2 } $$

$$X$$ has a binomial distribution with $$n = 20$$ and $$p=\displaystyle\frac { 1 }{ 2 } $$

$$\therefore P\left( X=x \right) =_{  }^{ n }{ { C }_{ x } }{ q }^{ n-x }{ p }^{ x }$$, where $$x=0,1,2,....n$$

$$=_{  }^{ 20 }{ { C }_{ x } }{ \left( \displaystyle\frac { 1 }{ 2 }  \right)  }^{ 20-x }\cdot { \left( \displaystyle\frac { 1 }{ 2 }  \right)  }^{ x }$$

$$=_{  }^{ 20 }{ { C }_{ x } }{ \left( \displaystyle\frac { 1 }{ 2 }  \right)  }^{ 20 }$$

$$P$$( at  least  12  questions  answered  correctly ) = $$P\left( X\ge 12 \right) $$

$$=P\left( X=12 \right) +P\left( X=13 \right) +...+P\left( X=20 \right) $$

$$=_{  }^{ 20 }{ { C }_{ 12 } }{ \left( \displaystyle\frac { 1 }{ 2 }  \right)  }^{ 20 }+_{  }^{ 20 }{ { C }_{ 13 } }{ \left( \displaystyle\frac { 1 }{ 2 }  \right)  }^{ 20 }+...+_{  }^{ 20 }{ { C }_{ 20 } }{ \left( \displaystyle\frac { 1 }{ 2 }  \right)  }^{ 20 }$$

$$={ \left( \displaystyle\frac { 1 }{ 2 }  \right)  }^{ 20 }\cdot \left[ _{  }^{ 20 }{ { C }_{ 12 } }+_{  }^{ 20 }{ { C }_{ 13 } }+...+_{  }^{ 20 }{ { C }_{ 20 } } \right] $$

Mathematics
RS Agarwal
Standard XII

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