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I have X, Y, and Z all as binary variables, values either 0 or 1. Y and Z are and got values of P(Y = 1), P(Z = 1), P(X = 1|Y = 1, Z = 1) , P(X = 1|Y = 1, Z = 0) and P(X = 1|Y = 0). here I need to compute P(X = 1)

I learned here I need to apply the concepts of law of total probability, conditional probability, and joint probability.

By reading through below link, I get to know how to calculate P(A| B, C, D)-

How can I calculate the conditional probability of several events?

I also got to know, this would be the formula - P(X = 1) = P(X = 1|Y = 1, Z = 1)P(Y = 1)P(Z = 1) + P(X = 1|Y = 1, Z = 0)P(Y = 1)P(Z = 0) + P(X = 1|Y = 0)P(Y = 0)

however, I haven't got any reference for this formula, do anyone has any reference for this or would be able to derive this?

ANKAN MAZUMDAR
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  • It might help you to explain to use what exactly you mean by "several events." After all, aren't you just describing a single event given by the conjunction of $(B,C,D)$? – whuber Sep 20 '23 at 19:50
  • As I mentioned I have gone through the link I added in my question, however that link doesn't have any reference how to calculate P(X=1) when other values are given – ANKAN MAZUMDAR Sep 22 '23 at 00:33
  • Consider the case of two variables $X$ and $Y$. Suppose you know the probabilities for $(X,Y)$ equal to $(0,0),$ $(1,0),$ $(0,1),$ and $(1,1).$ Notice that the event $X=1$ is the disjoint union of the events $(X,Y)=(1,0)$ and $(X,Y)=(1,1).$ Apply an appropriate probability axiom to obtain the answer. It's no different with more variables. – whuber Sep 22 '23 at 23:01

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