可能性がある means "there is a possibility that", "may". The clause in front describes like any noun: a verb or い-adjective in the plain form, a な-adjective with な or である, a noun with の or である.
。
There is a possibility that an earthquake will occur.
。
This medicine may cause side effects.
。
The data may have been deleted.
。
It is unlikely that he is the culprit.
可能性がある or かもしれない
is the everyday "maybe": the speaker's own guess. 可能性がある sounds measured and objective, and belongs to news, forecasts, doctors and reports. Said about your own small plans, it sounds stiff.
。
I might not be able to go tomorrow.
。
There is a possibility of heavy rain tomorrow.
How likely it is
可能性 is the subject, so , and go straight after が in place of ある.
。
Snow is likely tomorrow.
、。
The chance of success is low, but let's try.
は and も in place of が
denies the possibility outright, and
adds it to others already on the table.
。
There is no possibility of an accident.
。
There is also a possibility that the contract won't be renewed.
Lingaku