<s>[INST] how can calculate $E(X^Y|X+Y=1)$ let $X,Y$ are two independent random variables Bernoulli with probability $P$. how can calculate $E(X^Y|X+Y=1)$
 [/INST] Let's calculate all outcomes for the $X^Y$:


*

*$X=0$, $Y=0$ $\Rightarrow X^Y= 0^0 = 1$, $P=(1-p)^2$

*$X=0$, $Y=1$ $\Rightarrow X^Y= 0^1 = 0$, $P=p(1-p)$

*$X=1$, $Y=0$ $\Rightarrow X^Y= 1^0 = 1$, $P=p(1-p)$

*$X=1$, $Y=1$ $\Rightarrow X^Y= 1^1 = 1$, $P=p^2$
Condition $X+Y=1$ means we consider only the two
equally likely outcomes 2 and 3, that is, conditioned
on $X+Y = 1$, $X^Y$ is a Bernoulli random variable with
parameter $\frac{1}{2}$ and so $E[X^Y\mid X+Y = 1] = \frac{1}{2}$.
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