Feeling Lucky?

Suppose you have a biased coin which comes up heads with probability \(\frac{1}{3}\) and tails with probability \(\frac{2}{3}\). You then begin to toss the coin repeatedly, with heads worth 1 point and tails worth 2 points, and add up the points as you go along.

Let \(p(n)\) be the probability that at some stage you have accumulated precisely \(n\) points. (For example, \(p(1) = \frac{1}{3}, p(2) = \frac{2}{3} + (\frac{1}{3})^{2} = \frac{7}{9}\), etc..)

If \(\displaystyle\lim_{n\rightarrow \infty} p(n) = \dfrac{a}{b}\), where \(a\) and \(b\) are positive coprime integers. Find \(a + b\).

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