A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.
A die is thrown again and again until three sixes are obtained. Find the probability of obtaining the third six in the sixth throw of the die.
The probability of getting a six in a throw of die is $\frac{1}{6}$ and not getting a six is $\frac{5}{6}$.
Let $p=\frac{1}{6}$ and $q=\frac{5}{6}$
The probability that the 2 sixes come in the first five throws of the die is ${ }^{5} C_{2}\left(\frac{1}{6}\right)^{2}\left(\frac{5}{6}\right)^{3}=\frac{10 \times(5)^{3}}{(6)^{5}}$
$\therefore$ Probability that third six comes in the sixth throw $=\frac{10 \times(5)^{3}}{(6)^{5}} \times \frac{1}{6}$
$=\frac{10 \times 125}{(6)^{6}}$
$=\frac{10 \times 125}{46656}$
$=\frac{625}{23328}$
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