Confused about law of total variance

[Probabilist1]

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 machine is set to state A. If the result is 3 or 4, the machine is set to state B. If it is 5 or 6, the machine is set to C. Now in state A, the lottery amount is 500 with probability 0.2, 1000 with probability 0.5, 2000 with probability 0.3. In state B, the amount is 500 with probability 0.3, 1000 with probability 0.4, 2000 with probability 0.3. In state C, the amount is 500 with pr. 0.1, 1000 with pr. 0.3, 2000 with pr. 0.6. Determine the *variance* of the amount" (yes, it's a long question...)

Doing this directly (i.e. using probabilities (1/3)(0.2+0.3+0.1) for 500, etc.) I get E(amount)=1300 and Var(amount)=360000

However, if I try using the law of total variance by conditioning on the state, I get a different result. Letting S represent the lottery state and X the amount,

E(X|S)=1200 if S=A, 1150 if S=B, 1550 if S=C
Var(X|S)=310000 if S=A, 352500 if S=B, 322500 if S=C
E(E(X|S))=1300
Var(E(X|S))=31666.66
E(Var(X|S))=328166.66

From the law of total variance, we should have Var(X)=E(Var(X|S))+Var(E(X|S)) right? But that gives 31666.66+328166.66=359833.33 which is not 360000... am I doing a stupid calculation mistake somewhere??

 

[Jameson]

Hi Probabilist, (Wave)

Welcome to MHB!

I get the same thing when doing this the "direct way", as you put it. :)

Can you show your work for how you calculated Var(X|S)?

 

[Probabilist1]

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.