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# A Fair Die is Thrown Twenty Times. the Probability that on the Tenth Throw the Fourth Six Appears is(a) 20 C 10 × 5 6 6 20 - Mathematics

ConceptBernoulli Trials and Binomial Distribution

#### Question

A fair die is thrown twenty times. The probability that on the tenth throw the fourth six appears is

• $\frac{ ^{20}{}{C}_{10} \times 5^6}{6^{20}}$

• $\frac{120 \times 5^7}{6^{10}}$

• $\frac{84 \times 5^6}{6^{10}}$

• None of these

#### Solution

$\frac{84 \times 5^6}{6^{10}}$

$\text{ Let p be the probabilty of obtaining a six in a single throw of the die . Then } ,$
$p = \frac{1}{6}\text{ and }q = 1 - \frac{1}{6} = \frac{5}{6}$
$\text{ Obtaining a fourth six in the tenth throw of the die means that in the first nine throws }$
$\text{ there are 3 sixes and the fourth six is obtained in the tenth throw . Therefore, required probability }$
$= P(\text{ Getting 3 sixes in the first nine throws } ) P(\text{ Getting a six in the tenth throw } )$
$= \left( ^{9}{}{C}_3 \ p^3 q^{9 - 3} \right) p$
$=^{9}{}{C}_3 \left( \frac{1}{6} \right)^3 \left( \frac{5}{6} \right)^6 \times \frac{1}{6}$
$= \frac{84 \ x \ 5^6}{6^{10}}$

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Solution A Fair Die is Thrown Twenty Times. the Probability that on the Tenth Throw the Fourth Six Appears is(a) 20 C 10 × 5 6 6 20 Concept: Bernoulli Trials and Binomial Distribution.
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