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## Homework Statement

"Toss a coin repeatedly. Denote the event of getting heads or tails on the ##i##-th try by ##H_i## and ##T_i## respectively, where ##P(H_i)=p## and ##P(T_i)=1-p##, for some ##0\leq p \leq 1##. Now denote by ##E## the event of getting ##r## consecutive heads before ##s## consecutive tails. Find ##P(E|H_1)##, ##P(E|T_1)##, and ##P(E)##."

## Homework Equations

__Independence:__If ##A## and ##B## are events, then ##P(A\cap B)=P(A)P(B)##.

## The Attempt at a Solution

So far, I have denoted ##E## by ##E=(H_{j+1} \cap H_{j+2} \cap ...\cap H_{j+r} \cap T_{k+1} \cap ... \cap T_{k+s})## for some ##j\in ℕ## and for some ##k\in ℕ,\space k>j+r##. The major thing that concerns me at the moment, is the set of ##H_i,T_i## for ##1\leq i \leq j## and the set of ##H_m,T_m## for ##j+r+1\leq m \leq k##. That's really all I could come up with myself.

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