Probability of the hunter to shoot the target exactly 6 times

A shooter can hit a given target with probability $\dfrac{1}{4}$. She keeps firing a bullet at the target until she hits it successfully three times and then she stops firing. What is the probability that she fires exactly six bullets?

I am clueless it will be appreciated if any clue is provided without writing the complete solution.

The sixth shot must be the third hit. Two of the first five shots must be hits. So: $$P(X=2)\cdot \frac14=\begin{pmatrix} 5 \\ 2\end{pmatrix}\cdot \left(\frac14\right)^2\cdot \left(\frac34\right)^3\cdot \frac14=\frac{135}{2048}.$$
• $P(X=2)$ is the probability of two hits out of five shots. It is the binomial distribution: $P(X=x)=\begin{pmatrix} n \\ x\end{pmatrix}\cdot p^n\cdot (1-p)^{n-x}$. And $\frac14$ is the probability of a hit. The two probabilities are multiplied as the events are independent. Apr 11, 2018 at 17:40