Two teams \( A \) and \( B \) play a tournament. The first one to w...

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Two teams \( A \) and \( B \) play a tournament. The first one to win \( (n+1) \) games win the series. The probability
P that \( A \) wins a game is \( p \) and that \( B \) wins a game is \( q \)

W (no ties). Find the probability that \( A \) wins the series.
Hence or otherwise prove that \( \sum_{r=0}^{n}{ }^{n+1} C_{r} \cdot \frac{1}{2^{n+r}}=1 \).
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