24 -- A device is designed so that it randomly receives one of two letters A or B as input and passes through three stages. At each stage, the input letter is printed with probability $\frac{1}{r}$ without change, or it moves to the next stage changed to the other letter. If the probability of selecting letter A as input is 2 times that of letter B, with what probability is ``A'' printed by the device equal to the probability of the input letter being A? (1) $\dfrac{14}{23}$ (2) $\dfrac{21}{22}$ (3) $\dfrac{9}{41}$ (4) $\dfrac{17}{41}$ %% Page 7
\textbf{24 --} A device is designed so that it randomly receives one of two letters A or B as input and passes through three stages. At each stage, the input letter is printed with probability $\frac{1}{r}$ without change, or it moves to the next stage changed to the other letter. If the probability of selecting letter A as input is 2 times that of letter B, with what probability is ``A'' printed by the device equal to the probability of the input letter being A?
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(1) $\dfrac{14}{23}$ \hfill (2) $\dfrac{21}{22}$ \hfill (3) $\dfrac{9}{41}$ \hfill (4) $\dfrac{17}{41}$
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