3 miles to the fair

Discrete Mathematics Level pending

Harvey decides to go to the fair which is three miles from his house.

The problem is, every mile he has a \(\frac{1}{3}\) chance of changing direction, and going directly opposite whichever way he is currently going.

If he winds up back at his house, he gives up on going to the fair and finds other distractions.

What is the probability that he will never make it to the fair?

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