# How Many Four-Game Series?

In the finals of a rugby tournament, two teams play a best of 5 series. Each team has a probability of $$\frac{1}{2}$$ of winning the first game. For each subsequent game, the team that won the previous game has a $$\frac{7}{10}$$ chance of winning, while the other team has a $$\frac{3}{10}$$ chance of winning. If $$p$$ is the probability that the series lasts exactly 4 games, what is $$\lfloor 1000p \rfloor$$?

Details and assumptions

Greatest Integer Function: $$\lfloor x \rfloor: \mathbb{R} \rightarrow \mathbb{Z}$$ refers to the greatest integer less than or equal to $$x$$. For example $$\lfloor 2.3 \rfloor = 2$$ and $$\lfloor -3.4 \rfloor = -4$$.

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