# Question about spin states in Susskind's Quantum Mechanics

This is my first question on this site so please improve the question if needed. I am a beginner in Quantum Mechanics and so to study about it, I purchased the book "The Theoretical Minimum : Quantum Mechanics by Leonard Susskind" as an introduction to Quantum Mechanics. I am currently on lecture-2 which is about Quantum States and the author is talking about "spin states" of a system. I have completed "spin along z-axis" and "spin along x-axis" but I am having a doubt in "spin along y-axis". In the spin along y-axis section, the author uses "in" and "out" states labelled as $$|i\rangle$$ (in) and $$|o\rangle$$ (out). Then the author mentions that they are orthogonal states i.e. $$\langle i|o\rangle = 0$$ and vice-versa which mean that if the spin is in then it has $$0$$ probability to be out which I am clear with. Then the author uses statistical results of our experiments and writes : \begin{align*} \langle o|u\rangle \langle u|o\rangle &= \frac12 \tag{1}\\ \langle o|d\rangle \langle d|o \rangle &= \frac12 \tag{2}\\ \langle i|u\rangle \langle u|i \rangle &= \frac12 \tag{3}\\ \langle i|d \rangle \langle d|i \rangle &= \frac12 \tag{4}\\ \end{align*} where $$u$$ and $$d$$ are the up and down states

I am having a problem in understanding how the product of these "inner-products" equal $$\frac12$$

My attempt : I know that \begin{align*} \langle A|u\rangle \langle u|A\rangle &= P_u \\ \langle A|d \rangle \langle d|A \rangle &= P_d \end{align*} where $$P_u$$ and $$P_d$$ are the probabilities for measurements of up and down states and $$|A\rangle$$ is a generic state. So in the same way, $$\langle o|u\rangle \langle u|o\rangle$$ is the probability that the spin is up given that out is our generic state and here $$|o\rangle$$ plays the role of $$|A\rangle$$. The probability is $$\frac12$$ because we can choose between two states namely up and down and hence the probability is $$\frac12$$ and the same holds for the other $$3$$. Is my logic correct or is there something missing? Any help/hint is appreciated

$$|A>$$ is the state in which the spin was prepared. Susskind now does an experiment with "turning the apparatus". If you do this to measure $$|u>$$ (for example), then you get the probability $$P_u$$. If the prepared spin state $$|A>$$ is $$|o>$$, then you get your probability of 1/2. I think this is pretty much your understanding anyways, correct? In further chapters Susskind explains also what happens when the apparatus is turned around an arbitrary angle.