What is the Trinity?

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I was raised Jehovah's Witness and they do not believe in the Trinity at all. Now I do have a basic knowledge of how the Trinity works but it seems to me that their are variations on this. I was tauught that God, Jesus, and the Holy Spirit are all the same person, working in different capacities. I think that they are distictively seperate people and that in the Bible there are places when it says God, it is plural, all of them, a family, working together as a unit. What do you think?