I don't think we have enough information to deduce what would happen here when you have a short or burnt LED. There are a few different ways the string could be wired...
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Lets say you have three colors and you have them in a linear order (Y - G - R) over and over again down the chain. So the string is wired so you could blink the different colors independently. Then you would have three intertwined strands and the plug would house the blink controller.
Then we get into the construction of the bulbs them selves. In many of the newer incandescent bulbs there is a special wire that shorts across the given bulb so the string is still working.