## Let $X_1,X_2,…,X_n$ be i.i.d uniform random variables on $[0,1]$. Define $Y=max${$X_1,X_2,…,X_n$}. Find $E[X_1|Y]$

Let $$X_1,X_2,…,X_n$$ be i.i.d uniform random variables on $$[0,1]$$. Define $$Y=max$${$$X_1,X_2,…,X_n$$}. Find $$E[X_1|Y]$$.

My answer: $$P(X_i|Y)=P(X_j|Y)$$ for all $$i,j=1,2,…,n$$.
Since $$\sum_{i=n}^{n}P(X_i|Y)=1$$, I get $$P(X_i|Y)=1/n$$.
Then $$E[X_1|Y]=\int_{0}^{1}x_1\times{1\over n}dx_1={1\over 2n}$$ Is it correct for $$P(X_i|Y)=P(X_j|Y)$$ and $$\sum_{i=n}^{n}P(X_i|Y)=1$$?