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(B) Let $P(H) = \frac{1}{3}$ and $P(T) = \frac{2}{3}$.The experiment stops with heads if we get $HH$ or sequences like $HTHH, HTHTHH, \dots$ or $THH,

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$\frac{5}{21}$

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(B) Let $P(H) = \frac{1}{3}$ and $P(T) = \frac{2}{3}$.The experiment stops with heads if we get $HH$ or sequences like $HTHH, HTHTHH, \dots$ or $THH, THTHH, THTHTHH, \dots$.Case $1$: Sequences starting with $H$ and ending in $HH$ are $HH, HTHH, HTHTHH, \dots$.This is a geometric series with first term $a = P(H) \cdot P(H) = \frac{1}{3} \cdot \frac{1}{3} = \frac{1}{9}$ and common ratio $r = P(T) \cdot P(H) = \frac{2}{3} \cdot \frac{1}{3} = \frac{2}{9}$.Sum $= \frac{1/9}{1 - 2/9} = \frac{1/9}{7/9} = \frac{1}{7}$.Case $2$: Sequences starting with $T$ and ending in $HH$ are $THH, THTHH, THTHTHH, \dots$.This is a geometric series with first term $a = P(T) \cdot P(H) \cdot P(H) = \frac{2}{3} \cdot \frac{1}{3} \cdot \frac{1}{3} = \frac{2}{27}$ and common ratio $r = \frac{2}{9}$.Sum $= \frac{2/27}{1 - 2/9} = \frac{2/27}{7/9} = \frac{2}{27} \cdot \frac{9}{7} = \frac{2}{21}$.Total probability $= \frac{1}{7} + \frac{2}{21} = \frac{3+2}{21} = \frac{5}{21}$.

Consider an experiment of tossing a coin repeatedly until the outcomes of two consecutive tosses are the same. If the probability of a random toss resulting in a head is $\frac{1}{3}$,then the probability that the experiment stops with heads is:

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