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Grammar experts, another article question. These two sentences:

1. There is always a possibility that he might go back to Seattle.

2. This study raises the possbility that dieting is bad for health.

I thought that that-clause after a noun modifies the noun, changing its article to the definite one 'the'. The first sentence got an indefinite article even though it is modified by that-clause. Would anyone explain to me? Thanks so much!
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There may be a "rule" governing your question, but I am unaware of it. I'm sure I use them interchangeably.