Can one tell me if the treatment of the Mach-Zehnder interferometer in

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Can one tell me if the treatment of the Mach-Zehnder interferometer in the attachment is correct ?
 

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My problem is : Since one could reason in terms of possible path : if 2 path are possible, only X clicks, else both X and Y, but this would lead to an intensity of 25% for Y.
 

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