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the game restarts with $a$ being the second player; so $$p_n=p_{heads}( -p_{n- })+p_{tails}( -p_n)= rac ( -p_{n- })+ rac ( -p_n)$$ and $$p_n= rac{ -p_{n- }}{ }$$ or ( thanks @j.w.tanner ) $$p_n=\dfrac -\dfrac \left(-\dfrac ight)^n$$ the solution for "$n$ is even" case is the same up to heads $\leftrightarrow...
cohen marty cohen . k gold badges silver badges bronze badges $\endgroup$ $egingroup$ thanks, this does help. however, i'm looking for some intuition in terms of the interpretation of $lpha- $ being the number of heads observed. why is the distribution telling us the coin is either all heads or all tails...
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