I think that most people today would agree with you that the behavior of a single particle isn't deterministic. However, if you want me to state it clearly, let me first try to explain what it actually means.
Assume that you are sitting at a table with a friend, but there are three chairs. Your eyes are closed, so you can't see in which chair your friend is sitting. Since cheating is not allowed, once you open your eyes the probability of finding your friend to the right or to the left is about equal (superposition of eigenstates). Furthermore, say that you are sitting in a plane flying at high speed. The probability of finding your friend in a certain place thus propagates in a deterministic fashion through space. However, once you open your eyes you will see your friend sitting in only one of the two chairs. The probability of finding him in the other suddenly becomes zero. (The measurement 'pushes' the system into one eigenstate.) And you will always find him in the same chair even if you repeat the measurement by quickly closing your eyes and then opening them again.
This is very much how a simple quantum mechanical system works. It can, for instance, be an electron with its spin pointing up or down. Before you measure the spin, there is a certain probability of finding it in either state, but if you repeat the measurement, you will always see the same thing that you measured before. However, your friend clearly knows in which chair he is sitting. The probability you assign to each possibility represents your imperfect knowledge of the world. The question about randomness is thus the following. Does the electron know which way its spin points, or does it only (randomly) make up its mind about it once you decide to actually measure its spin? Of course, the electron probably isn't very bright, but it can have an internal (hidden) variable which determines what it is 'really' doing.
It may seem difficult to distinguish the two cases, but there are actually strong indications, such as Bell's inequalities, that 'hidden' variables do not exist. Without going into detail, the basic idea of Bell's work is to say that you can't have it all. If you believe one thing, you may have to give up something else.