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Suppose in a fictional penalty shoot-out, you get 3 shoots in a row. Your team is 1 point behind. According to a statistical study, there is an 80% chance that you will score a goal in a penalty kick. So, what is the probability of winning the game? Note 1- It should not be a tie. Note 2- When you score 2 goals in a row, you are declared the winner, and not allowed to perform the third kick. Note 3- Remember, its a fictional game. Do not correlate it with the actual game. So, keep these two things in mind- i) only YOU are going to take all the three penalties, not any other team member. ii) your opponent team does not get a penalty shoot-out. They are done playing!

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Probability you win $= 1- $ probability that you did not win

We did not win if we score at most $1$ goal.

$$1-\left(0.2^3+\begin{pmatrix} 3 \\ 1 \end{pmatrix} 0.2^2(0.8)\right)$$

Alternatively, we can directly compute

$$\left(0.8^3+\begin{pmatrix} 3 \\ 2 \end{pmatrix} 0.8^2(0.2)\right)$$

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    look at my **Note 2**. So, isn't it wrong using 0.2^3 or 0.8^3? I think it should be 0.2^2 or 0.8^2. – 2017-01-27
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    if you failed in your first $2$ attempts, do you get a third attempts? if so you have $0.2^3$ right? – 2017-01-27
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    anyway, you can imagine that there is always the third kick, where the score doesn't matter as long as you have minimal of $2$ goals. – 2017-01-27