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LFM Pure
LFM Pure and Mechanics
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UFM Additional Further Pure
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Coordinate Geometry
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Problem Text
A biased coin, with a probability $p$ of coming up heads and a probability $q=1-p$ of coming up tails, is tossed repeatedly. Let $A$ be the event that the first run of $r$ successive heads occurs before the first run of $s$ successive tails. If $H$ is the even that on the first toss the coin comes up heads and $T$ is the event that it comes up tails, show that \begin{alignat*}{1} \mathrm{P}(A|H) & =p^{\alpha}+(1-p^{\alpha})\mathrm{P}(A|T),\\ \mathrm{P}(A|T) & =(1-q^{\beta})\mathrm{P}(A|H), \end{alignat*} where $\alpha$ and $\beta$ are to be determined. Use these two equations to find $\mathrm{P}(A|H),$ $\mathrm{P}(A|T),$ and hence $\mathrm{P}(A).$
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