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What difference does 'a' and 'the' make to the two sentence below?

  1. There is a possibility that the source of water will be polluted
  2. There is the possibility that the source of water will be polluted
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In context, either is valid.

The standard difference between "a" and "the" is that "a" refers to one of many possible instances, while "the" refers to a specific instance. Like, "I saw a dog": I saw some dog, it could have been any dog. "I saw the dog": I saw the one specific dog that is presently under discussion.

But here, either article logically means the same thing. There could be many possible ways that the water could be polluted, so you could say "a possibility". Or you could be thinking of the possibility that the water is polluted versus the possibility that it is not, in which case there is only one possibility, and it's "the". Either way, the sentence means the same thing.

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