Oops... I was sure I double-checked all my work, but it seems I made a calculation error when calculating E(Var(X|S)), correct value is 328333.33 which works out. Very sorry, I don't make that kind of mistake usually.
Ok, so I got this question on an exam some time ago and I still don't understand why I didn't get it (I can't remember the exact question, but this is very similar):
"A lottery winning amount is determined in the following manner: first a die is thrown. If the result is 1 or 2, the lottery...