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I'm reading through Jayne's Probability Theory: The Logic of Science and he mentions that given a set of propositions {$A_1A_2...A_n$} that are mutually exclusive given information $B$ (meaning, any two of them can't be true simultaneously given $B$), they can be expressed as

$$p(A_iA_j|B) = p(A_i|B)\delta_{ij}$$

What is the $\delta_{ij}$?

I would think that the probability $p(A_iA_j|B) =0$ so I'm not sure what the point of the right-hand side even is.

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  • $\begingroup$ $p(A_iA_j|B)$ need not be $0$ when $i=j$ i.e. when $A_i$ and $A_j$ are the same event $\endgroup$ – Henry Jun 14 '18 at 0:37
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This is the "Kronecker delta". The value of $\delta_{ij}$ is $1$ when $i=j$ and $0$ otherwise.

It's not specific to probability theory.

In your example you are right to say that the right hand side is $0$ - except when $i=j$.

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