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Is there a closed form for getting three in a row given n hands, as opposed to just three hands? I've solved this problem using dynamic programming, I'm just wondering if you can get a closed formula based on p and n.
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Here is a
simple recursion.
This old thread describes the Markov chain solution which involves matrix multiplication, as well as several approximate closed form methods, including a very accurate one from Feller which depends on finding the dominant root of a polynomial.
If you can find all the roots of the polynomial that results from taking the z-transform of the difference equation, then in theory you could do a partial fraction expansion and write a closed-form solution in terms of exponentials.