I have a question regarding finding the probability of measuring a specific particle in a mixed ensemble. Say you have a mixed ensemble containing 30% +z particles and 70% -z.

Then the probability of taking a measurement and observing a +z particle is 30%. This makes perfect sense given the ratios of particles. But what I am confused about is the state could be represented by $$|\psi\rangle=\frac{3}{10}|+z\rangle+\frac{7}{10}|-z\rangle$$ using $|\psi\rangle=P_1|+z\rangle+P_2|-z\rangle$. But this is not a normalized function. I was under the impression that to take the probability of a quantum state the state must first be normalized. But if we normalize this state and then take the probability then we get the wrong probabilities. I am very confused about where my misunderstanding is coming from. Any help would be appreciated!

Cheers