3 - Venn Diagram Question Finding some serious problem with the famous "Venn Diagram" probability problems. I have "worked out" three such problems associated with a diagram and am wondering if you could pick out flaws in my reasoning!
For conditions A, B, C, sample size 500, here is my distribution.

I. Finding probability of selecting one of the 500 that meets the condition: A but not B.
Here I just used shading to find the numerical answer. I shaded in all of B and then counted remaining of A.

This means 210 - 122 - 52 + 97 = 133. Probability 133/500!
II. Finding probability of selecting one of the 500 that meets the condition: C AND B but not A.
Super easy here. There are only 83 that meet C and B at all. Then 52 that meet C, B, and A. So 83 - 52 = 31. 31/500 probability!
III. Finding probability of selecting one of the 500 that meets the condition: (C or A)'.

I just shaded all members of C or A then counted what was left. 258 - 122 - 52 - 83 = 1. 1/500 probability!
Is there anything wrong with my logic here? Is there a better way to do this aside from shading? Are there explicit formulas, for example, for each of these?
 A: The  number of people who satisfy $A$ and not $B$ is not calculated correctly. There are $210$ in $A$. Of these, $122$ are in the part in common between $A$ and $B$. The difference is $88$, giving probability $88/500$.
The calculation in the second problem is correct.
For the third problem, it is useful to first count the number of people that satisfy "$C$ or $A$".  If we add $210$ and $216$, we double count the $97$ people who are in both $C$ and $A$. So the number of people in $C$ or $A$ is $210+216-97$. That's $329$. We want the number of people in $(C\cup A)'$, so we subtract $329$ from $500$. 
The result is $171$. A calculation could be made along your lines. One has to take account of the people who are outside all of $A,B,C$.
Remark: The numbers in the diagram are placed in ways that make confusion likely.  Note that the diagram divides the world into $8$ pieces. If you are going to solve several problems about this diagram, it will be useful to write in each piece the number of people in that piece.  As things stand, only the central piece is labelled with a number ($52$) that indicates the number of people in that piece.  
