Hmm, well, I think "big possibility" is more likely to be used in speech. "High possibility" is pretty common in both speech and writing. If you have any doubts about whether something "sounds right", you should see if there is an English collocation guide available in your local library. This kind of reference lists entries (words) and describe words that commonly appear together with them. I'm making up an example here but: ex. "Expectation" high, big, low, poor 1. She had high ~s of her eldest daughter (== She had high expectations of her eldest daughter). 这个句子我不知道怎么翻译到中文，对不起。“父母的 要求，期待 很大/高”？

Possibility can alo be used when there is a certain chance for something to happen: "I believe there is a high possibility that tomorrow will rain."There is a low possibility that tomorrow will rain." or "There is a high possibility that you will succeed." and in case you are not sure then you would say "There is a low possibility that you will succeed." I hope this helped, have a nice day.

"Possibility" has many meanings. Depending on the actual meaning you can use either "big" or "high".