Why does the following procedure fail:

mkdir -p /home/user/chroot/{bin,lib}
sudo mount --bind /bin /home/user/chroot/bin
sudo mount --bind /lib /home/user/chroot/lib
sudo chroot /home/user/chroot
I am getting the following error:

chroot: failed to run command `/bin/bash': No such file or directory
The thing is /home/user/chroot/bin/bash exists since I bind mounted the directory.

In addition, there are procedures which bind mount proc and sys and then chroot into a livecd or something at which point proc and sys are available.

So basically, why doesn't the chroot know about the bind mounted /bin/bash? Why is this procedure I'm doing incorrect? I figure for what I'm doing I don't need proc or sys and that I would get other errors if not including them int he chroot was causing the problem.