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$ ?