1

I found both expressions.

  1. There is a possibility that he will come.

  2. There is the possibility that it was deliberate.

What's the difference?

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2

Using 'a' indicates you are talking about one of several potential things (such as possibilities). Using 'the' indicates you are talking about a specific thing (or possibility). The two words have similar but not identical meanings in your example sentences.

For example, picking 'a coin' out of a pile of coins means any single coin you happen to pick out of the pile of coins, while picking 'the coin' out of the same pile means you have picked a specific coin out of that pile. Usually if there is no difference between coins, 'a coin' is sufficient, while if you use 'the coin' it implies 'the coin' is different in some way (and often more important) than all the other coins.

Below, I will expand your example sentences and include new sentences with close or equivalent meaning to the example sentences to help you understand the difference between using 'a' or 'the' in them:

1) There is a possibility that he will come.

He may come or he may not come.

2) There is the possibility that he will come.

He may or may not come, but his coming is important to whatever we are discussing. 

3) There is a possibility that it was deliberate.

It may have been deliberate, but that is just one of several possibilities.

4) There is the possibility that it was deliberate.

It may have been deliberate, and if it was deliberate that fact is important somehow.
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0

Here is a simple way to think about it:

If you use 'a', it means that you're talking about any given instance of an object or event.

E.g, 'I saw a cow'. It sounds like you're passively commenting on an event for the first time.

If you use 'the', it means that you're talking about a SPECIFIC instance of an object/event, and the specific object is relevant to whatever's going on. E.g. 'I saw the cow'. It sounds like this cow has already been mentioned somewhere in context, and so is significant to the situation, and the listener in question already knew about the cow.

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'The' is used in one of 3 contexts:

1) When there is only one of a thing in a given context.

E.g. On a ship there is normally one captain, the captain. Or, "Within the universe there are many stars."

2) When you're mentioning something for the 2nd time.

E.g. "I was watching a bird eat. Then the bird started to fly."

3) When the context implies that you're talking about something specifically.

E.g. "Please press the button to open the doors."

Using the with the concepts 'possibility', 'chance', et cetera, makes the phrase somewhat more definite and specific. 'The' perhaps implies it's a bigger deal, as in the only possibility, or that it's been discussed for some time.

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