I have a quantum state of 3-qubits : $|\psi\rangle = \frac{1}{2}|0\rangle (|a\rangle \otimes|b\rangle+|b\rangle\otimes |a\rangle) + \frac{1}{2}|1\rangle (|a\rangle \otimes|b\rangle - |b\rangle\otimes |a\rangle)$.
What's the probability that it is in the $|0\rangle$ state if I measure it (one of the qubits) in the $|0\rangle,|1\rangle$ basis?
I'm expecting that it will be $|\frac{1}{2}(|a\rangle \otimes|b\rangle+|b\rangle\otimes |a\rangle) |^2$ but I'm not sure how that is equal to $\frac{1}{2}+\frac{1}{2}|\langle a|b\rangle|$ (see Page 7 here).
We know $|a\rangle,|b\rangle, |0\rangle, |1\rangle \in \Bbb C^2$
Could someone please explain.
I think this is more of a math question so I am asking it here instead of Physics SE.