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I am sure that most people are aware of the infinite monkey theorem. If not use http://en.wikipedia.org/wiki/Infinite_monkey_theorem as a reference.

By using this theorem, could one say that the the first billion decimal digits of pi (in order) almost certainly would show up somewhere in the decimal digits of phi? Of course where this phenomenon would occur would start at some unimaginably enormous number.

I assume this would be true. Since phi is irrational, the digits in its decimal expansion have no pattern, so essentially they are random.

To take this one step further, could one say that the said pattern of the first billion digits of pi, would occur within the decimal expansion of phi an infinite number of times?