Regions A, B, C, J and K represent ponds. Logs leave pond A and float down flumes (represented by arrows) to eventually end up in pond B or pond C. On leaving a pond, the logs are equally likely to use any available exit flume. Logs can only float in the direction the arrow is pointing. What is the probability that a log in pond A will end up in pond B? Express your answer as a common fraction.
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+26387 Regions A, B, C, J and K represent ponds.
Logs leave pond A and float down flumes (represented by arrows) to eventually end up in pond B or pond C.
On leaving a pond, the logs are equally likely to use any available exit flume.
Logs can only float in the direction the arrow is pointing.
What is the probability that a log in pond A will end up in pond B?
Express your answer as a common fraction.
\(\begin{array}{|rcll|} \hline \mathbf{C} &=& \mathbf{(AJ-JC) + (AJ-CJ) + (AK-KC) + AC} \\\\ C &=& \left(\dfrac{1}{3}*\dfrac{1}{3}\right) +\left(\dfrac{1}{3}*\dfrac{1}{3}\right) + \left(\dfrac{1}{3}*\dfrac{1}{2}\right) + \dfrac{1}{3} \\\\ C &=& \dfrac{1}{9} + \dfrac{1}{9} + \dfrac{1}{6} + \dfrac{1}{3} \\\\ C &=& \dfrac{2}{9} + \dfrac{1}{6} + \dfrac{1}{3} \\\\ C &=& \dfrac{2}{9} + \dfrac{1}{6} + \dfrac{3}{9} \\\\ C &=& \dfrac{5}{9} + \dfrac{1}{6} \\\\ C &=& \dfrac{5*6+1*9}{9*6} \\\\ C &=& \dfrac{39}{54} \\\\ \mathbf{C} &=& \mathbf{\dfrac{13}{18}} \\ \hline \end{array} \begin{array}{|rcll|} \hline \mathbf{B} &=& \mathbf{(AJ-JB) + (AK-KB)} \\\\ B &=& \left(\dfrac{1}{3}*\dfrac{1}{3}\right) + \left(\dfrac{1}{3}*\dfrac{1}{2}\right) \\\\ B &=& \dfrac{1}{9} + \dfrac{1}{6} \\\\ B &=& \dfrac{6+9}{9*6} \\\\ B &=& \dfrac{15}{54} \\\\ \mathbf{B} &=& \mathbf{\dfrac{5}{18}} \\ \hline \end{array}\)
