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I am looking at a problem in a text book and it asks "how many independent values in a joint probability distribution for eight boolean nodes, assuming no conditional independence relations are known to hold among them"

The answer it gives is 2^8 - 1 = 255

I cant work out why the minus one is in there though. Can anyone help?

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    $\begingroup$ The last entry is not independent because the total of all the entries must sum to one. So one entry is fixed by 1 minus the sum of the other 255. $\endgroup$
    – mandata
    Commented May 7, 2015 at 3:30

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With 8 Boolean variables, the joint has $2^8-1 = 255$ independent entries. In general, the joint must specify $2^n-1$ numbers (where n is the number of variables). The -1 comes from the fact that the sum of those numbers must add up to 1 because the probability of all possible outcomes must sum to 1.

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