We have had this debate since time immemorial, 'prove that God exists'. It depends on the person who is asking the question and his/her views about the topic.
Look at a glass that has water up to the midway. A pessimistic would say that the glass is half empty, an optimist would say that the glass is half full. Similarly, there would be many instances in real life that the believers would attribute to the presence of Almighty and an equal number who ask for scientific reasons would say it would have happened anyway.
There are few personal circumstances from which I have learned to believe the existence of God but they would have to be taken at face value. Instead, let's see some common things in life. We see near hills or small mountains, huge boulders that are sitting still without any support. At the base we would see few houses. The rocks can tumble down anytime but they don't. One of the best examples in the rock at Madan Mahal fort, Jabalpur. A geologist would put many theories about shape, weight etc but those who believe in God would take His name.
There are so many accidents (excluding drunk/rash/negligent driving) that happen, every day before a journey most of us say a small prayer (even I do), why most of us come home intact and few suffer. This is an act of God.
Some might remember brick buildings or concrete brick walls having perfectly healthy plants fully green with roots hanging out. People tend their gardens,use manure and good water but their sapling die out; yet, these plants take life and hang on for a while bearing flowers and fruits. A horticulturalist would have theories about adaptation, capillary action, atmospheric water etc but a believer would say it's destiny of the Divine.
At the last, I would put down our own existence. Any idea how man found his way to Earth ? Anthropologists would have many theories running back for many million years but its finally due to the existence of God.