# A Fair Die is Tossed Eight Times. the Probability that a Third Six is Observed in the Eighth Throw Isa) 7 C 2 × 5 5 6 7 (B) 7 C 2 × 5 5 6 8 (C) 7 C 2 × 5 5 6 6(D) None of These - Mathematics

#### 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

#### Solution

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

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

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