#### Question

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

##### Options

\[\frac{ ^{20}{}{C}_{10} \times 5^6}{6^{20}}\]

\[\frac{120 \times 5^7}{6^{10}}\]

\[\frac{84 \times 5^6}{6^{10}}\]

None of these

#### Solution

\[\frac{84 \times 5^6}{6^{10}}\]

\[\text{ Let p be the probabilty of obtaining a six in a single throw of the die . Then } , \]

\[p = \frac{1}{6}\text{ and }q = 1 - \frac{1}{6} = \frac{5}{6}\]

\[\text{ Obtaining a fourth six in the tenth throw of the die means that in the first nine throws } \]

\[\text{ there are 3 sixes and the fourth six is obtained in the tenth throw . Therefore, required probability } \]

\[ = P(\text{ Getting 3 sixes in the first nine throws } ) P(\text{ Getting a six in the tenth throw } )\]

\[ = \left( ^{9}{}{C}_3 \ p^3 q^{9 - 3} \right) p\]

\[ =^{9}{}{C}_3 \left( \frac{1}{6} \right)^3 \left( \frac{5}{6} \right)^6 \times \frac{1}{6}\]

\[ = \frac{84 \ x \ 5^6}{6^{10}}\]