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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