This is how I understand your question:

If the average rain per month is X and for the first 2 weeks there have already have been X rain, does this mean that the rain probability for the next 2 weeks would be higher or lower?

There is no easy answer. There is no monthly rain quota where you used up your monthly rain early. One will have to analyze the reasons why there was more rain than usual.

Still, globally, I would be inclined to say that if the first part of the month was not normal, there is somewhat higher than average probability that the second won't be either. Weather patterns come from global cycles and global weather systems - and it in this case something is off. For example the European weather is highly influenced by the current state of the Azores High and the Icelandic Low. If these are off for some reason, it usually takes some time before they gradually return to normal.