By definition, if the probability of an event increases, the probability of the event "happening" increases. Is not an event something that happens?
But I think I know what you're trying to get at. Here's my analysis.
First, let E be the event in question. I think you are trying to express something to the tune of:
P(E) < P(E|P(E))
Taken at face value, P(E) is the probability of the event. P(E|P(E)) is the probability of the event given that we know the probability of the event. This is a situation that is somewhat analogous to stock market dynamics - the more certain the market is that a stock will reach a low price before a certain time, the more they will sell, thus increasing the probability that the low price will be reached before the specified time.
The problem with this formalism is that P(E) is not a random variable. It is a number. E is a random variable. It is therefore associated with some distribution. Perhaps the event has a 30% chance of happening and a 70% chance of not happening. Then drawing values from E will take on one value 70% of the time and another value 30% of the time (however you want to represent happening and not happening - 0 and 1 are always nice).
Now let's suppose P(E) was a random variable. Then the above equation is not expressable. P(E) is not subject to the < operator because it does not represent a single value.
Therefore what I think you are looking for is a bit more Baysian.
Suppose there is a concept Ev(E) which is the evidence that E will occur. This is a proposition, not a random variable. As Ev(E) goes to infinity, P(E) goes to 1. Let P(E) = 1/2 represents the proposition that the you have equal evidence for and against the event happening. You have a certain amount of evidence for it, as represented by Ev(E), and your evidence determines the state of your belief. What you are suggesting is that knowing the value of Ev(E) increases it. Let's make that another proposition:
K = 'I know the value of Ev(E)'
Now consider the statement
Ev(E) < Ev(E|K)
P(E) < P(E|K).
What I think you are asking about is really a statement about the change in the amount of evidence you have and its effect on your belief. In the Baysian framework, probabilities are equivalent to beliefs, which is why concepts like this can be addressed a bit more... naturally.
To answer your question, the name for
"awareness of the probability of an event increasing the probability of it
is equivalent to
"awareness of your belief of an event (happening) increasing your belief of that it
This phenomenon is called:
Second guessing your beliefs.
Or more accurately in this case, as you explicity use the word increasing:
Becoming more confident in your beliefs.
Perhaps the take away here is that probabilies are not real. They require infinite data to construct and would only be "true: if you could roll back time to collect the exact same data again an again. And if you believe in a deterministic world, then you wouldn't need a probability in the first place. You'd just know. If you believe in a random world and you have the power to roll it back over and over again, and then relive the situation, you get to choose the probability.
That silliness is one of the reasons why probabilities are best understood as represent a state of belief. And if your awareness of your own belief changes it, well that's just questioning yourself and thus reassessing your beliefs.
If you believe in a true probability of an event happening, then your awareness of that probability could not change it, as your awareness is already factored into the true probability of that event happening.
If you're feeling mathy, give this a try: