Question

A fair die is tossed eight times. The probability that a third six is observed in the eighth throw is

Options

• \[\frac{^{7}{}{C}_2 \times 5^5}{6^7}\]

   

• \[\frac{^{7}{}{C}_2 \times 5^5}{6^8}\]

• \[\frac{^{7}{}{C}_2 \times 5^5}{6^6}\]

   

• None of these

   

Answer 1

\[\frac{^{7}{}{C}_2 \times 5^5}{6^8}\]

\[\text{ Let p be the probabilty of obtaining a six in a single throw of the die . Then , }  \]
\[p = \frac{1}{6}\text{ and } q = 1 - \frac{1}{6} = \frac{5}{6}\]
\[\text{ Obtaining a third six in the eighth throw of the die means that in first seven throws }  \]
\[\text{ there are 2 sixes and the third six is obtained in the eighth throw . Therefore, }  \]
\[\text{ required probability} \]
\[ = P(\text{ Getting 2 sixes in the first seven throws} ) P( \text{ Getting six in the eighth throw } )\]
\[ = \left(^{7}{}{C}_2 p^2 q^{7 - 2} \right) p\]
\[ = ^{7}{}{C}_2 \left( \frac{1}{6} \right)^2 \left( \frac{5}{6} \right)^5 \times \frac{1}{6}\]
\[ = \frac{^{7}{}{C}_2 \ x \ 5^5}{6^8}\]