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Part 1

A baby is learning to walk with the assistance of its living room couch. The baby starts at the couch and at each given time the baby will make a decision. It will either take a brave step forward, stand in place not knowing what to do, or fearfully take a step back towards the couch with probabilities 0.20.2, 0.50.5, and 0.30.3 respectively. The baby will never go behind the couch (so when at the couch the baby has probability 0.20.2 of moving forward and probability 0.80.8 of staying at the couch).

If you were to observe this baby for an extremely long amount of time, what proportion of the time would the baby be at the couch?