A precipitate will appear if the ion product is greater than Ksp, or if the amount of substance added to solution is higher than the molar solubility.
Ion product is the "Q" of a dissolved substance. It tells you how much "stuff" you have dissolved. Ksp tells you the maximum "stuff" you can have dissolved, and if you pass it, some will have to precipitate.
Molar solubility tells you how many moles of a substance will dissolve in a liter of solution. So if molar solubility is 1, 1 mole of that substance can effectively dissolve in solution without any precipitate. But if you add 1.1 moles to 1 liter, you'll get some precipitate.
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