Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
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Mark the Correct Alternative in the Following Question: the Probability of Guessing Correctly at Least 8 Out of 10 Answers of a True False Type Examination is - Mathematics

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Question

Mark the correct alternative in the following question:

The probability of guessing correctly at least 8 out of 10 answers of a true false type examination is

Options
  • \[\frac{7}{64}\]

  • \[\frac{7}{128}\]

  • \[\frac{45}{1024} \]

  • \[\frac{7}{41}\]

Solution

\[\text{ We have,}  \]

\[p = \text{ probabiltiy of guessing the answer of a true false correctly } = \frac{1}{2} \text{ and } \]

\[q = \text{ probabiltiy of guessing the answer of a true false incorrectly }  = 1 - p = 1 - \frac{1}{2} = \frac{1}{2}\]

\[\text{ Let X denote a success of guessing the answer correctly . Then, } \]

\[\text{ X follows the binomial distribution with parameters n = 10 and } p = \frac{1}{2}\]

\[ \therefore P\left( X = r \right) = ^{10}{}{C}_r p^r q^\left( 10 - r \right) = ^{10}{}{C}_r \left( \frac{1}{2} \right)^r \left( \frac{1}{2} \right)^\left( 10 - r \right) = ^{10}{}{C}_r \left( \frac{1}{2} \right)^{10} = \frac{^{10}{}{C}_r}{2^{10}}\]

\[\text{ Now } , \]

\[\text{ Required probability } = P\left( X \geq 8 \right)\]

\[ = P\left( X = 8 \right) + P\left( X = 9 \right) + P\left( X = 10 \right)\]

\[ = \frac{^{10}{}{C}_8}{2^{10}} + \frac{^{10}{}{C}_9}{2^{10}} + \frac{^{10}{}{C}_{10}}{2^{10}}\]

\[ = \frac{45 + 10 + 1}{2^{10}}\]

\[ = \frac{56}{1024}\]

\[ = \frac{7}{128}\]

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APPEARS IN

 RD Sharma Solution for Mathematics for Class 12 (Set of 2 Volume) (2018 (Latest))
Chapter 33: Binomial Distribution
MCQ | Q: 30 | Page no. 30
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Mark the Correct Alternative in the Following Question: the Probability of Guessing Correctly at Least 8 Out of 10 Answers of a True False Type Examination is Concept: Bernoulli Trials and Binomial Distribution.
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