The Road Not Taken...

Suppose you have \(24\) identical toothpicks and arrange them so that they form an outline of a \(3\) by \(3\) grid of \(9\) identical squares.

Starting at the lower left corner, we can trace out \(20\) distinct paths to the top right corner by going either up or to the right, one toothpick at a time. (Each of these paths involve precisely \(6\) toothpicks.)

If we remove one of the \(24\) toothpicks at random, what is the expected number of these \(20\) distinct paths that will remain intact?

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