Mass m in Ring of Radius r: Probability of Particle Existing is Zero

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Consider a mass m confined to a ring of radius r. The potential everywhere on the ring is zero. In the ml = +3 state, identify the points on the ring where the probability of the particle existing is zero.

I was thinking that every point would be zero, because it's a wave, not a particle. It cannot exist at anyone point, basically. Am I thinking about this wrong?
 
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I am guessing you are supposed to think about what standing wave function(s) would be allowed, given a ring of certain size and a particle of certain momentum. In all cases a standing wave will have nodes, and the probability of the particle being at a node is zero. If I remember correctly, you get the probability function by squaring the wave function and normalizing.

You are right, sort of, about a wave not really existing at any given point, but that is a macroscopic analogy and doesn't quite translate to the quantum world. In quantum, wave functions for particles do have places where they are "more likely to exist."