Cohen-Tannoudji on mutually exclusive (?) events

[terra]

I was looking at what Cohen-Tannudji has to say on compatibility of observables.
Assumptions: $A,B$ are operators such that $[A,B]=0$ and we denote $|a_i \,b_j\rangle$ to be states for which $A | a_i \, b_j \rangle= a_i | a_i \, b_j \rangle$, $B | a_i \, b_j \rangle= b_j | a_i \, b_j \rangle$.
We start with the state $|\psi \rangle= \sum_{i,j} c_{i,j} | a_i \, b_j \rangle$.
The whole discussion starts with the following:
"The probability for finding $a_1$ is $P(a_1)= \sum_{j} |c_{1,j}|^2$." (Page 232 in a 1977 edition.)
Have I forgotten something fundamental? I thought that the amplitudes $\langle a_1 \, b_{j'} | \psi \rangle$ and $\langle a_1 \, b_j | \psi \rangle$ are mutually exclusive for $j' \neq j$, so that according to quantum rules for probability
$$P(a_1)= \big| \sum_j \langle a_1 \, b_{j} | \psi \rangle \big|^2= \sum_j \sum_{j'} \langle a_1 \, b_j | \psi \rangle \langle \psi | a_1 \, b_{j'} \rangle. $$
I see no reason as to why $j'= j$ should hold.
My apologies for the slightly dull question, but I'm a bit lost.

 

[mathman]

I'm completely ignorant of the physics here. However when dealing with probabilities, mutually exclusive means the probability of both happening is 0.

 

[terra]

I'm completely ignorant of the physics here. However when dealing with probabilities, mutually exclusive means the probability of both happening is 0.

Thanks for the reply, yeah.
Let $a,b$ be exclusive events. For classical physical things $P(a\mathrm{ \, or \, }b)= P(a) + P(b)$. In quantum physics, however, we have $A(a\mathrm{ \, or \, }b)= A(a) + A(b)$ where $A \in \mathbb{C}$ is a 'probability amplitude' so that $P(x)= A(X) A^*(X)$, star denoting a complex conjugate, for some event $X$. This definition will bring so called interference terms when compared with the classical case (they disappear in the classical limit).
In my case, $| a_i \, b_j \rangle$ are vectors.. some physical states, in fact, that have well-defined (=certain) values for some observables $A,B$ (so, a physical state always has some value or a distribution of values for both). We have different states for different $i$ and $j$. A term $\langle a_1 \, b_j | \psi \rangle$ is actually just such an amplitude for the state $| \psi \rangle$ having the value $a_1$ for $A$ and the value $b_j$ for $B$. I'm trying to determine the total probability to find that the value of $A$ for $|\psi \rangle$ is $a_1$. As I see it, $|\psi \rangle$ can have that value while having $b_1,b_2,b_3,...$ for $B$.