You have read that the universe is 15 billion years old. This number is the best guess that scientists have right now. You are correct in saying that no one was alive to see the universe begin, so we need to try to figure out the age of the universe by indirect means. These indirect methods are, unfortunately, not very good, and until recently most people thought the universe was between 10 and 20 billion years old, but they really couldn't say with any certainty what the actual value is.
One method which has been used to estimate the age of the universe is to look for the piece of the universe that is the farthest from us. If we know how fast it is moving away from us and how far away from us, we can figure out how long ago it was that the piece of the universe and the earth were at the same point. This point corresponds to the big bang which was the origin of the universe.
As a problem for you, you might want to think about how we would do this calculation. If we know how far an object is from us and how fast it is moving away from us, we only have to make a simple calculation to figure out how long ago the object was at the same point as us.
Lastly, I should note that there is still some uncertainty about the age of the universe. The reason for this uncertainty is that not all scientists agree that the universe has been expanding at the same rate since the big bang. If this rate changes, then our calculation of the age of the universe will be flawed, because we are assuming that the farthest part of the universe has been moving away from us at the same speed since the beginning of the universe.
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