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I am thinking about one question, the result seems weird, but I could not figure out where I went wrong.

I have two random Bernoulli variables $X$ and $Y$.

$P(Y=1)$

$=P(Y=1, X=1)+P(Y=1,X=0)$

$=P(Y=1|X=1)P(X=1)+P(Y=1|X=0)P(X=0)$

$=P(Y=1|X=1)P(X=1)+P(Y=1|X=0)(1 - P(X=1))$

$=P(Y=1|X=0)+[P(Y=1|X=1)-P(Y=1|X=0)]P(X=1)$

It seems that a probabilities of one Bernoulli random variable is always a linear function of another Bernoulli random variable no matter of what! This does not make sense to me. Could you please let me know where I went wrong?

Thanks.

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  • $\begingroup$ The probabilities are linearly linked but that does not imply that the variables themselves are no? $\endgroup$
    – user83346
    Feb 18 '17 at 7:45
  • $\begingroup$ Because you only compute the probabilities at two different values of the other variable, no matter how you might imagine the relationship to be, it will correspond precisely to one conditional probability at X=0 and another conditional probability at X=1; those two values can always be written as a linear function. $\endgroup$
    – Glen_b
    Feb 19 '17 at 3:17
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You didn't go wrong. Your equations are correct. Possibly you are letting the fact that you are working with probabilities make this seem more mysterious than it is? What you've discovered is a variant of the idea that any quantity is a linear combination of itself and some other quantity: X = X + 0(Y) = 1*X*(a + (1 - a)) = etc. for any X and Y

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