Important Note: For "layman" read "next to zero understanding of QM mathematics"
I am reading Quantum Mechanics: The Theoretical Minimum. In Chapter 2 on Quantum States, the following is presented:
$$|A\rangle = a_u |u\rangle + a_d |d\rangle$$
$$a_u = \langle u|A\rangle$$ $$a_d = \langle d|A\rangle$$
where $|A\rangle$ is a generic state that can represent the state of a spin, and $a_u$, $a_d$ are components of $|A\rangle$ along the basis vectors $|u\rangle$ and $|d\rangle$.
The book then states that the probability that the spin will be up if measured along the z axis is $a_u^* a_u$, i.e., the square of the magnitude of the probability amplitude.
But then it presents:
$$P_u = \langle A|u\rangle \langle u|A\rangle$$
where $P_u$ is the probability of the spin being measured as up.
$$P_u = \langle u|A\rangle \langle u|A\rangle$$
given what was stated earlier?