# 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.

Question:

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.

Solution:

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 $=\frac{10 \times(5)^{3}}{(6)^{5}} \times \frac{1}{6}$

$=\frac{10 \times 125}{(6)^{6}}$

$=\frac{10 \times 125}{46656}$

$=\frac{625}{23328}$