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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.

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2 Answers 2

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Consider what situation she would fire six bullets. She would have to hit successfully for the 3rd time on her 6th bullet. As a result, in between, she would have to hit successfully exactly 2 times; it may help to utilize the binomial distribution to calculate the probability of this component.

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The sixth shot must be the third hit. Two of the first five shots must be hits. So: $$P(X=2)\cdot \frac14=\begin{pmatrix} 5 \\ 2\end{pmatrix}\cdot \left(\frac14\right)^2\cdot \left(\frac34\right)^3\cdot \frac14=\frac{135}{2048}.$$

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  • $\begingroup$ Can you please elaborate your process? $\endgroup$ Apr 11, 2018 at 17:29
  • $\begingroup$ $P(X=2)$ 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. $\endgroup$
    – farruhota
    Apr 11, 2018 at 17:40

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