Do you believe in ghosts?

This is a serious question.
I have just been reading something put up by someone I have never met but I would have thought was a very down-to-earth sort of person. I still think she is that sort of person. She used to be in the police force. The police are not generally known for fantasies of the sort she was describing.
I won't go into details. It's her story, not mine.
I can tell you another story - or two. 
Recently I went to the funeral of a man in his late 90's. If anyone was a down-to-earth sort of man then he was. He was quiet, calm, analytical to a point which drove those around him "up the wall". He was a scientist. He dealt in facts, not fiction.
I knew him most of my life and he always seemed to be the same - apart from once. There was a blip in his life that he could never explain. It bothered him.
I might never have found out about it except that I happened to arrive at his home one Saturday and he was standing outside looking down the street.
     "Did someone just go past you?" he asked me and his wife as we came across the street. He was looking agitated.
     "No."
     "No," I echoed.
     "I know I didn't leave the house unlocked. I had to use the keys to get in."
I should explain here that it took two keys to enter their house. Security was tight because of the work he was doing at the time.
He then told  us that he had been out too. When he came home he unlocked the back door and went in. As he went up the passage someone had come out of the room he  used as his study.When challenged he had just nodded at this man and walked past him as if nothing was wrong.
     "I felt him walk past me."
He had immediately gone to the front door - and had to unlock that. 
When we went into the house he and his wife checked every window and tested all the locks. There was no sign of a break-in.
She said he must have been imagining it - but she also knew that imagination of that sort was completely out of character for him.
He went into his study. It was one of those  ultra-tidy sort of places, in keeping with his personality. There were papers scattered across the desk.
He never did find an explanation.
I don't know what the answer is either. All I can think of is that he had some sort of "episode" and had done it all himself while momentarily mentally disturbed.  He never experienced a similar episode but it bothered him for the rest of his life.
Other people I have known have told me of other incidents that they have not been able to fully explain too. They have usually been people with a great deal of imagination or people I have considered to be prone to believing things they are told without questioning them at all.
I think I have imagination too - probably too much of it. It is probably that which caused both my brother and I to do something which was/is out of character. When we were in our teens my maternal grandmother died. We were, very reluctantly, living with her at the time - something we both deeply resented. She was not a nice woman at all, indeed a dreadful one in many ways. 
She went to lie down after lunch one afternoon and didn't wake up. My brother found her. It was a shocking experience for a sixteen year old boy. I was not there at the time or I would no doubt have been the one to check why she has not woken from her usual afternoon nap. 
We children did not attend the funeral - such as it was. Even the Senior Cat did not attend it - at our mother's request. Our grandmother had not had friends. By then she had alienated everyone around her. 
About a week later my brother and I were alone in the house. In the middle of the afternoon I felt an urge to leave the house. There was no reason to do it. I came out of the room I was sleeping in and found my brother had just come out of the room he was sleeping in.
    "Let's get out of here," he told me.
We went outside. A moment later, in the room my grandmother had died in, there was a small bang - as if something had hit the floor. There was nobody in there.  
When we finally went back into the house I found the powder compact she had used on the floor. It seemed to have been flung down with great force because it had broken. There was no explanation for it although logic tells me that it must have been knocked to the edge by my mother and then fallen. 
But why did we both feel the need to leave the house at that point - and how could we hear a powder compact falling on a carpet from outside the house?
I am sure there have to be reasonable explanations for these things - and many others.
I just don't know what they are. Any suggestions?