Question
A fair die is tossed eight times. The probability that a third six is observed on the eighth throw is :
A.
$${}^7{C_2}\,\frac{{{5^5}}}{{{6^8}}}$$
B.
$${}^7{C_3}\,\frac{{{5^3}}}{{{6^8}}}$$
C.
$${}^7{C_6}\,\frac{{{5^6}}}{{{6^8}}}$$
D.
none of these
Answer :
$${}^7{C_2}\,\frac{{{5^5}}}{{{6^8}}}$$
Solution :
The required event occurs if two sixes are observed in the first seven throws and a six is observed on the eighth throw. If $$p$$ is the probability that a six shows on the die, the number of throws $$n$$ is $$7$$, and $$X$$ is the number of times a six is observed, then $$X \sim B\left( {7,\,p} \right).$$
Therefore the required probability equals $$P\left( {X = 2} \right)$$ times the probability of getting a six on the eighth throw, i.e., it equals
$$\eqalign{
& = \left( {{}^7{C_2}\,{p^2}{q^5}} \right)\left( p \right) \cr
& = \left( {{}^7{C_2}} \right){\left( {\frac{1}{6}} \right)^2}{\left( {\frac{5}{6}} \right)^5}\left( {\frac{1}{6}} \right) \cr
& = \frac{{{}^7{C_2}\left( {{5^5}} \right)}}{{{6^8}}} \cr} $$