Starting at value 0 , the fortune of an investor increases per week by \(\$ 200\) with probability \(3 / 8\), remains constant with probability \(3 / 8\), and decreases by \(\$ 200\) with probability \(2 / 8\). The weekly increments of the investor's fortune are assumed to be independent. The investor stops the 'game' as soon as he has made a total fortune of \(\$ 2000\) or a loss of \(\$ 1000\), whichever occurs first.

By using suitable martingales and applying the optional stopping theorem, determine (1) the probability \(p_{2000}\) that the investor finishes the 'game' with a profit of \(\$ 2000\), (2) the probability \(p_{-1000}\) that the investor finishes the 'game' with a loss of \(\$ 1000\), (3) the mean duration \(E(N)\) of the 'game.'