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

#### 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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