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counting and probability intermediate

Problem

Regions and represent ponds. Logs leave pond and float down flumes (represented by arrows) to eventually end up in pond or pond . 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 will end up in pond ? Express your answer as a common fraction.

problem
Solution
There are two paths from A to B: A to K to B, and A to J to B. The probability that a log will go from A to K to B is the probability that it will choose the middle flume initially times the probability that it will choose the flume to the right given that it chose the middle flume initially: . Similarly, the probability that the log will go from A to J to B is . In total, the probability that the log reaches B is .
Final answer
\frac{5}{18}