(i) Players \( A \) and \( B \) alternately toss a biased coin, with \( A \) going first. \( A \... VIDEO
(i) Players \( A \) and \( B \) alternately toss a biased coin, with \( A \) going first. \( A \) wins if \( A \) tosses a Tail before \( B \) tosses a Head; otherwise \( B \) wins. If the probability of a head is \( p \), find the value of \( p \) for which the game is fair to both players.
(ii) The entries in a two-by-two determinant \( \left|\begin{array}{ll}a & b \\ c & d\end{array}\right| \) are integers that are chosen randomly and independently, and for each entry, the probability that the entry is odd is \( p \). If the probability that the value of the determinant is even is \( 1 / 2 \), then find the value of \( p \).
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